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CalculusCalculus (MindTap Course List)Let A be the area under the graph of an increasing continuous function f from a to h , and let L n and R n be the approximations to A with n subintervals using left and right endpoints, respectively. (a) How are A , L n , and R n related? (b) Show that R n − L n = b − a n [ f ( b ) − f ( a ) ] Then draw a diagram to illustrate this equation by showing that the n rectangles representing R n - L n can be reassembled to form a single rectangle whose area is the right side of the equation. (c) Deduce that R n − A < b − a n [ f ( b ) − f ( a ) ]BuyFind*arrow_forward*

8th Edition

James Stewart

Publisher: Cengage Learning

ISBN: 9781285740621

Chapter 4.1, Problem 27E

Textbook Problem

Let A be the area under the graph of an increasing continuous function *f* from *a* to *h*, and let *L _{n}* and

(a) How are *A*, *L _{n}*, and

(b) Show that

Then draw a diagram to illustrate this equation by showing that the *n* rectangles representing *R _{n} *-

*(c)* Deduce that

Calculus (MindTap Course List)

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Ch. 4.1 - a By reading values from the given graph of f, use...Ch. 4.1 - a Use six rectangles to find estimates of each...Ch. 4.1 - a Estimate the area under the graph of f(x)=1/x...Ch. 4.1 - a Estimate the area under the graph of f(x)=sinx...Ch. 4.1 - a Estimate the area under the graph of f(x)=1+x2...Ch. 4.1 - a Graph the function f(x)=1/(1+x2)2x2 b Estimate...Ch. 4.1 - Evaluate the upper and lower sums for f(x)=2+sinx,...Ch. 4.1 - Evaluate the upper and lower sums for...Ch. 4.1 - With a programmable calculator or a computer, it...Ch. 4.1 - With a programmable calculator or a computer, it...

Ch. 4.1 - Some computer algebra systems have commands that...Ch. 4.1 - a If f(x)=x/(x+2),1x4, use the commands discussed...Ch. 4.1 - The speed of a runner increased steadily during...Ch. 4.1 - The table shows speedometer readings at 10-second...Ch. 4.1 - Oil leaked from a tank at a rate of r(t) liters...Ch. 4.1 - When we estimate distances from velocity data, it...Ch. 4.1 - The velocity graph of a braking car is shown. Use...Ch. 4.1 - The velocity graph of a car accelerating from rest...Ch. 4.1 - In someone infected with measles, the virus level...Ch. 4.1 - The table shows the number of people per day who...Ch. 4.1 - Use Definition 2 to find an expression for the...Ch. 4.1 - Use Definition 2 to find an expression for the...Ch. 4.1 - Use Definition 2 to find an expression for the...Ch. 4.1 - Determine a region whose area is equal to the...Ch. 4.1 - Determine a region whose area is equal to the...Ch. 4.1 - a Use Definition 2 to find an expression for the...Ch. 4.1 - Let A be the area under the graph of an increasing...Ch. 4.1 - If A is the area under the curve y=sinx from 0 to...Ch. 4.1 - a Express the area under the curve y=x5 from 0 to...Ch. 4.1 - a Express the area under the curve y=x4+5x2+x from...Ch. 4.1 - Find the exact area under the cosine curve y=cosx...Ch. 4.1 - a Let An be the area of a polygon with n equal...Ch. 4.2 - Evaluate the Riemann sum for f(x)=x1,6x4, with...Ch. 4.2 - If f(x)=cosx0x3/4 evaluate the Riemann sum with n...Ch. 4.2 - If f(x)=x24,0x3, find the Riemann sum with n = 6,...Ch. 4.2 - a Find the Riemann sum for f(x)=1/x,1x2, with four...Ch. 4.2 - The graph of a function f is given. Estimate...Ch. 4.2 - The graph of g is shown. Estimate 24g(x)dx with...Ch. 4.2 - A table of values of an increasing function f is...Ch. 4.2 - The table gives the values of a function obtained...Ch. 4.2 - Use the Midpoint Rule with the given value of n to...Ch. 4.2 - Use the Midpoint Rule with the given value of n to...Ch. 4.2 - Use the Midpoint Rule with the given value of n to...Ch. 4.2 - Use the Midpoint Rule with the given value of n to...Ch. 4.2 - If you have a CAS that evaluates midpoint...Ch. 4.2 - With a programmable calculator or computer see the...Ch. 4.2 - Use a calculator or computer to make a table of...Ch. 4.2 - Use a calculator or computer to make a table of...Ch. 4.2 - Express the limit as a definite integral on the...Ch. 4.2 - Express the limit as a definite integral on the...Ch. 4.2 - Express the limit as a definite integral on the...Ch. 4.2 - Express the limit as a definite integral on the...Ch. 4.2 - Use the form of the definition of the integral...Ch. 4.2 - Use the form of the definition of the integral...Ch. 4.2 - Use the form of the definition of the integral...Ch. 4.2 - Use the form of the definition of the integral...Ch. 4.2 - Use the form of the definition of the integral...Ch. 4.2 - a Find an approximation to the integral 04(x23x)dx...Ch. 4.2 - Prove that abxdx=b2a22Ch. 4.2 - Prove that abx2dx=b3a33Ch. 4.2 - Express the integral as a limit of Riemann sums....Ch. 4.2 - Express the integral as a limit of Riemann sums....Ch. 4.2 - Express the integral as a limit of sums. Then...Ch. 4.2 - Express the integral as a limit of sums. Then...Ch. 4.2 - The graph of f is shown. Evaluate each integral by...Ch. 4.2 - The graph of g consists of two straight fines and...Ch. 4.2 - Evaluate the integral by interpreting it in terms...Ch. 4.2 - Evaluate the integral by interpreting it in terms...Ch. 4.2 - Evaluate the integral by interpreting it in terms...Ch. 4.2 - Evaluate the integral by interpreting it in terms...Ch. 4.2 - Evaluate the integral by interpreting it in terms...Ch. 4.2 - Evaluate the integral by interpreting it in terms...Ch. 4.2 - Evaluate 111+x4dxCh. 4.2 - Given that 0sin4xdx=38, what is 0sin4d?Ch. 4.2 - In Example 4.1.2 we showed that 01x2dx=13. Use...Ch. 4.2 - Use the properties of integrals and the result of...Ch. 4.2 - Use the results of Exercises 27 and 28 and the...Ch. 4.2 - Use the result of Exercise 27 and the fact that...Ch. 4.2 - Write as a single integral in the form abf(x)dx:...Ch. 4.2 - If 28f(x)dx=7.3 and 24f(x)dx=5.9, find 48f(x)dx.Ch. 4.2 - If 09f(x)dx=37 and 09g(x)dx=16, find...Ch. 4.2 - Find 05f(x)dx if f(x)={3forx3xforx3Ch. 4.2 - For the function / whose graph is shown, list the...Ch. 4.2 - If F(x)=2xf(t)dt, where f is the function whose...Ch. 4.2 - Each of the regions A, B, and C bounded by the...Ch. 4.2 - Suppose / has absolute minimum value m and...Ch. 4.2 - Use the properties of integrals to verify the...Ch. 4.2 - Use the properties of integrals to verify the...Ch. 4.2 - Use the properties of integrals to verify the...Ch. 4.2 - Use the properties of integrals to verify the...Ch. 4.2 - Use Property 8 of integrals to estimate the value...Ch. 4.2 - Use Property 8 of integrals to estimate the value...Ch. 4.2 - Use Property 8 of integrals to estimate the value...Ch. 4.2 - Use Property 8 of integrals to estimate the value...Ch. 4.2 - Use Property 8 of integrals to estimate the value...Ch. 4.2 - Use Property 8 of integrals to estimate the value...Ch. 4.2 - Use properties of integrals, together with...Ch. 4.2 - Use properties of integrals, together with...Ch. 4.2 - Which of the integrals 12xdx,121/xdx, and 12xdx...Ch. 4.2 - Which one of the integrals...Ch. 4.2 - Prove Property 3 of integrals....Ch. 4.2 - a If f is continuous on a, b, show that...Ch. 4.2 - Let f(x)=0 if x is any rational number and f(x)=1...Ch. 4.2 - Let f(0)=0 and f(x)=1/x if 0x1. Show that f is not...Ch. 4.2 - Express the limit as a definite intergal....Ch. 4.2 - Express the limit as a definite intergal....Ch. 4.2 - Find 12x2dx. Hint: Choose xi* to be the geometric...Ch. 4.3 - Explain exactly what is meant by the statement...Ch. 4.3 - Let g(x)=0xf(t)dt, where f is the function whose...Ch. 4.3 - Let g(x)=0xf(t)dt, where f is the function whose...Ch. 4.3 - Let g(x)=0xf(t)dt where f is the function whose...Ch. 4.3 - Sketch the area represented by g(x). Then find...Ch. 4.3 - Sketch the area represented by g(x). Then find...Ch. 4.3 - Use Part 1 of the Fundamental Theorem of Calculus...Ch. 4.3 - Use Part 1 of the Fundamental Theorem of Calculus...Ch. 4.3 - Use Part 1 of the Fundamental Theorem of Calculus...Ch. 4.3 - Use Part 1 of the Fundamental Theorem of Calculus...Ch. 4.3 - Use Part 1 of the Fundamental Theorem of Calculus...Ch. 4.3 - Use Part 1 of the Fundamental Theorem of Calculus...Ch. 4.3 - Use Part 1 of the Fundamental Theorem of Calculus...Ch. 4.3 - Use Part 1 of the Fundamental Theorem of Calculus...Ch. 4.3 - Use Part 1 of the Fundamental Theorem of Calculus...Ch. 4.3 - Use Part 1 of the Fundamental Theorem of Calculus...Ch. 4.3 - Use Part 1 of the Fundamental Theorem of Calculus...Ch. 4.3 - Use Part 1 of the Fundamental Theorem of Calculus...Ch. 4.3 - Evaluate the integral. 13(x2+2x4)dxCh. 4.3 - Evaluate the integral. 11x100dxCh. 4.3 - Evaluate the integral. 02(45t334t2+25t)dtCh. 4.3 - Evaluate the integral. 01(18v3+16v7)dvCh. 4.3 - Evaluate the integral. 19xdxCh. 4.3 - Evaluate the integral. 18x2/3dxCh. 4.3 - Evaluate the integral. /6sindCh. 4.3 - Evaluate the integral. 55dxCh. 4.3 - Evaluate the integral. 01(u+2)(u3)duCh. 4.3 - Evaluate the integral. 04(4t)tdtCh. 4.3 - Evaluate the integral. 142+x2xdxCh. 4.3 - Evaluate the integral. 12(3u2)(u+1)duCh. 4.3 - Evaluate the integral. /6/2csctcottdtCh. 4.3 - Evaluate the integral. /4/3csc2dCh. 4.3 - Evaluate the integral. 01(1+r)3drCh. 4.3 - Evaluate the integral. 12s2+1s2dsCh. 4.3 - Evaluate the integral. 12v5+3v6v4dvCh. 4.3 - Evaluate the integral. 1183zdzCh. 4.3 - Evaluate the integral....Ch. 4.3 - Evaluate the integral....Ch. 4.3 - Sketch the region enclosed by the given curves and...Ch. 4.3 - Sketch the region enclosed by the given curves and...Ch. 4.3 - Sketch the region enclosed by the given curves and...Ch. 4.3 - Sketch the region enclosed by the given curves and...Ch. 4.3 - Use a graph to give a rough estimate of the area...Ch. 4.3 - Use a graph to give a rough estimate of the area...Ch. 4.3 - Use a graph to give a rough estimate of the area...Ch. 4.3 - Use a graph to give a rough estimate of the area...Ch. 4.3 - Evaluate the integral and interpret it as a...Ch. 4.3 - Evaluate the integral and interpret it as a...Ch. 4.3 - What is wrong with the equation? 21x4dx=x33]21=38Ch. 4.3 - What is wrong with the equation? 124x3dx=2x2]12=32Ch. 4.3 - What is wrong with the equation?...Ch. 4.3 - What is wrong with the equation? 0sec2xdx=tanx]0=0Ch. 4.3 - Find the derivative of the function....Ch. 4.3 - Find the derivative of the function....Ch. 4.3 - Find the derivative of the function....Ch. 4.3 - Find the derivative of the function....Ch. 4.3 - Let F(x)=xcosttdt. Find an equation of the tangent...Ch. 4.3 - If f(x)=0x(1t2)cos2tdt, on what interval is f...Ch. 4.3 - On what interval is the curve y=0xt2t2+t+2dt...Ch. 4.3 - Let F(x)=1xf(t)dt, where f is the function whose...Ch. 4.3 - If f(1)=12, f is continuous, and 14f(x)dx=17, what...Ch. 4.3 - If f(x)=0sinx1+t2dt and g(y)=3yf(x)dx, find g(/6).Ch. 4.3 - The Fresnel function S was defined in Example 3...Ch. 4.3 - The sine integral function Si(x)=0xsinttdt is...Ch. 4.3 - Let g(x)=0xf(t)dt where f is the function whose...Ch. 4.3 - Let g(x)=0xf(t)dt where f is the function whose...Ch. 4.3 - Evaluate the limit by first recognizing the sum as...Ch. 4.3 - Evaluate the limit by first recognizing the sum as...Ch. 4.3 - Justify 3 for the case h0.Ch. 4.3 - If f is continuous and g and h are differentiable...Ch. 4.3 - a Show that 11+x31+x3forx0 b Show that...Ch. 4.3 - a Show that cos(x2)cosxfor0x1 b Deduce that...Ch. 4.3 - Show that 0510x2x4+x2+1dx0.1 by comparing the...Ch. 4.3 - Let f(x)={0ifx0xif0x12xif1x20ifx2 and...Ch. 4.3 - Find a function f and a number a such that...Ch. 4.3 - Suppose h is a function such that...Ch. 4.3 - A manufacturing company owns a major piece of...Ch. 4.3 - A high-tech company purchases a new computing...Ch. 4.3 - The following exercises are intended only for...Ch. 4.3 - The following exercises are intended only for...Ch. 4.3 - The following exercises are intended only for...Ch. 4.3 - The following exercises are intended only for...Ch. 4.3 - The following exercises are intended only for...Ch. 4.3 - The following exercises are intended only for...Ch. 4.4 - Verify by differentiation that the formula is...Ch. 4.4 - Verify by differentiation that the formula is...Ch. 4.4 - Verify by differentiation that the formula is...Ch. 4.4 - Verify by differentiation that the formula is...Ch. 4.4 - Find the general indefinite integral....Ch. 4.4 - Find the general indefinite integral. x54dxCh. 4.4 - Find the general indefinite integral....Ch. 4.4 - Find the general indefinite integral....Ch. 4.4 - Find the general indefinite integral....Ch. 4.4 - Find the general indefinite integral. t(t2+3t+2)dtCh. 4.4 - Find the general indefinite integral. 1+x+xxdxCh. 4.4 - Find the general indefinite integral. (u2+1+1u2)duCh. 4.4 - Find the general indefinite integral. (2+tan2)dCh. 4.4 - Find the general indefinite integral....Ch. 4.4 - Find the general indefinite integral....Ch. 4.4 - Find the general indefinite integral. sin2xsinxdxCh. 4.4 - Find the general indefinite integral. Illustrate...Ch. 4.4 - Find the general indefinite integral. Illustrate...Ch. 4.4 - Evaluate the integral. 23(x23)dxCh. 4.4 - Evaluate the integral. 12(4x33x2+2x)dxCh. 4.4 - Evaluate the integral. 20(12t4+14t3t)dtCh. 4.4 - Evaluate the integral. 03(1+6w210w4)dwCh. 4.4 - Evaluate the integral. 02(2x3)(4x2+1)dxCh. 4.4 - Evaluate the integral. 11t(1t)2dtCh. 4.4 - Evaluate the integral. 0(4sin3cos)dCh. 4.4 - Evaluate the integral. 12(1x24x3)dxCh. 4.4 - Evaluate the integral. 14(4+6uu)duCh. 4.4 - Evaluate the integral. 12(21p2)2dpCh. 4.4 - Evaluate the integral. 145xdxCh. 4.4 - Evaluate the integral. 18(2w2w3)dwCh. 4.4 - Evaluate the integral. 14t(1+t)dtCh. 4.4 - Evaluate the integral. 0/4sectandCh. 4.4 - Evaluate the integral. 0/41+cos2cos2dCh. 4.4 - Evaluate the integral. 0/3sin+sintan2sec2dCh. 4.4 - Evaluate the integral. 182+tt23dtCh. 4.4 - Evaluate the integral. 064u(uu3)duCh. 4.4 - Evaluate the integral. 01(x54+x45)dxCh. 4.4 - Evaluate the integral. 01(1+x2)3dxCh. 4.4 - Evaluate the integral. 25|x3|dxCh. 4.4 - Evaluate the integral. 02|2x1|dxCh. 4.4 - Evaluate the integral. 12(x2|x|)dxCh. 4.4 - Evaluate the integral. 03/2|sinx|dxCh. 4.4 - Use a graph to estimate the x-intercepts of the...Ch. 4.4 - Use a graph to estimate the x-intercepts of the...Ch. 4.4 - The area of the region that lies to the right of...Ch. 4.4 - The boundaries of the shaded region in the figure...Ch. 4.4 - If w(t) is the rate of growth of a child in pounds...Ch. 4.4 - The current in a wire is defined as the derivative...Ch. 4.4 - If oil leaks from a tank at a rate of r(t) gallons...Ch. 4.4 - A honeybee population starts with 100 bees and...Ch. 4.4 - In Section 3.7 we defined the marginal revenue...Ch. 4.4 - If f(x) is the slope of a trail at a distance of x...Ch. 4.4 - If x is measured in meters and f(x) is measured in...Ch. 4.4 - If the units for x are feet and the units for ax...Ch. 4.4 - The velocity function in meters per second is...Ch. 4.4 - The velocity function in meters per second is...Ch. 4.4 - The acceleration function in m/s2 and the initial...Ch. 4.4 - The acceleration function in m/s2 and the initial...Ch. 4.4 - The linear density of a rod of length 4 m is given...Ch. 4.4 - Water flows from the bottom of a storage tank at a...Ch. 4.4 - The velocity of a car was read from its...Ch. 4.4 - Suppose that a volcano is erupting and readings of...Ch. 4.4 - Lake Lanier in Georgia, USA, is a reservoir...Ch. 4.4 - Water flows into and out of a storage tank. A...Ch. 4.4 - The graph of the acceleration a(t) of a car...Ch. 4.4 - Shown is the graph of traffic on an Internet...Ch. 4.4 - The following graph shows the power consumption in...Ch. 4.4 - On May 7, 1992, the space shuttle Endeavour was...Ch. 4.4 - The following exercises are intended only for...Ch. 4.4 - The following exercises are intended only for...Ch. 4.4 - The following exercises are intended only for...Ch. 4.4 - The following exercises are intended only for...Ch. 4.4 - The following exercises are intended only for...Ch. 4.4 - The area labeled B is three times the area labeled...Ch. 4.5 - Evaluate the integral by making the given...Ch. 4.5 - Evaluate the integral by making the given...Ch. 4.5 - Evaluate the integral by making the given...Ch. 4.5 - Evaluate the integral by making the given...Ch. 4.5 - Evaluate the integral by making the given...Ch. 4.5 - Evaluate the integral by making the given...Ch. 4.5 - Evaluate the indefinite integral. x1x2dxCh. 4.5 - Evaluate the indefinite integral. x2sin(x3)dxCh. 4.5 - Evaluate the indefinite integral. (12x)9dxCh. 4.5 - Evaluate the indefinite integral. sin1+costdtCh. 4.5 - Evaluate the indefinite integral. sin(2/3)dCh. 4.5 - Evaluate the indefinite integral. sec22dCh. 4.5 - Evaluate the indefinite integral. sec3ttan3tdtCh. 4.5 - Evaluate the indefinite integral. y2(4y3)2/3dyCh. 4.5 - Evaluate the indefinite integral. cos(1+5t)dtCh. 4.5 - Evaluate the indefinite integral. sinxxdxCh. 4.5 - Evaluate the indefinite integral. sec2tan3dCh. 4.5 - Evaluate the indefinite integral. sinxsin(cosx)dxCh. 4.5 - Evaluate the indefinite integral. (x2+1)(x3+3x)4dxCh. 4.5 - Evaluate the indefinite integral. xx+2dxCh. 4.5 - Evaluate the indefinite integral. a+b23ax+bx3dxCh. 4.5 - Evaluate the indefinite integral. cos(/x)x2dxCh. 4.5 - Evaluate the indefinite integral. z21+z33dzCh. 4.5 - Evaluate the indefinite integral. dtcos2t1+tantCh. 4.5 - Evaluate the indefinite integral. cotxcsc2xdxCh. 4.5 - Evaluate the indefinite integral. sec2xtan2xdxCh. 4.5 - Evaluate the indefinite integral. sec3xtanxdxCh. 4.5 - Evaluate the indefinite integral. x22+xdxCh. 4.5 - Evaluate the indefinite integral. x(2x+5)8dxCh. 4.5 - Evaluate the indefinite integral. x3x2+1dxCh. 4.5 - Evaluate the indefinite integral. Illustrate and...Ch. 4.5 - Evaluate the indefinite integral. Illustrate and...Ch. 4.5 - Evaluate the indefinite integral. Illustrate and...Ch. 4.5 - Evaluate the indefinite integral. Illustrate and...Ch. 4.5 - Evaluate the definite integral. 01cos(t/2)dtCh. 4.5 - Evaluate the definite integral. 01(3t1)50dtCh. 4.5 - Evaluate the definite integral. 011+7x3dxCh. 4.5 - Evaluate the definite integral. 0xcos(x2)dxCh. 4.5 - Evaluate the integral. 0/6sintcos2tdtCh. 4.5 - Evaluate the definite integral. /32/3csc2(12t)dtCh. 4.5 - Evaluate the definite integral. /4/4(x3+x4tanx)dxCh. 4.5 - Evaluate the definite integral. cosxsin(sinx)dxCh. 4.5 - Evaluate the definite integral. 013dx(1+2x)23Ch. 4.5 - Evaluate the definite integral. 0axa2x2dxCh. 4.5 - Evaluate the definite integral. 0axx2+a2dx(a0)Ch. 4.5 - Evaluate the definite integral. /3/3x4sinxdxCh. 4.5 - Evaluate the definite integral. 12xx1dxCh. 4.5 - Evaluate the definite integral. 04x1+2xdxCh. 4.5 - Evaluate the definite integral. 1/21cos(x2)x3dxCh. 4.5 - Evaluate the definite integral. 0T/2sin(2t/T)dtCh. 4.5 - Evaluate the definite integral. 01dx(1+x)4Ch. 4.5 - Verify that f(x)=sinx3 is an odd function and use...Ch. 4.5 - Use a graph to give a rough estimate of the area...Ch. 4.5 - Use a graph to give a rough estimate of the area...Ch. 4.5 - Evaluate 22(x+3)4x2dx by writing it as a sum of...Ch. 4.5 - Evaluate 01x1x4dx by making a substitution and...Ch. 4.5 - Breathing is cyclic and a full respiratory cycle...Ch. 4.5 - A model for the basal metabolism rate, in kcal/h,...Ch. 4.5 - If f is continuous and 04f(x)dx=10, find...Ch. 4.5 - If f is continuous and 09f(x)dx=4, find...Ch. 4.5 - If f is continuous function on , prove that...Ch. 4.5 - If f is continuous function on , prove that...Ch. 4.5 - If a and b are positive numbers, show that...Ch. 4.5 - If f is continuous on [0,], use the substitution...Ch. 4.5 - If f is continuous, prove that...Ch. 4.5 - Use Exercise 65 to evaluate 0/2cos2xdx and...Ch. 4.5 - The following exercise are intended only for these...Ch. 4.5 - The following exercise are intended only for these...Ch. 4.5 - The following exercise are intended only for these...Ch. 4.5 - The following exercise are intended only for these...Ch. 4.5 - The following exercise are intended only for these...Ch. 4.5 - The following exercise are intended only for these...Ch. 4.5 - The following exercise are intended only for these...Ch. 4.5 - The following exercise are intended only for these...Ch. 4.5 - The following exercise are intended only for these...Ch. 4.5 - The following exercise are intended only for these...Ch. 4.5 - The following exercise are intended only for these...Ch. 4.5 - The following exercise are intended only for these...Ch. 4.5 - The following exercise are intended only for these...Ch. 4.5 - The following exercise are intended only for these...Ch. 4.5 - The following exercise are intended only for these...Ch. 4.5 - The following exercise are intended only for these...Ch. 4.5 - The following exercise are intended only for these...Ch. 4.5 - The following exercise are intended only for these...Ch. 4.5 - Use Exercise 64 to evaluate the integral...Ch. 4.R - a Write an expression for a Riemann sum of a...Ch. 4.R - a Write the definition of the definite integral of...Ch. 4.R - State the Midpoint Rule.Ch. 4.R - State both parts of the Fundamental Theorem of...Ch. 4.R - a State the Net Change Theorem. b If r(t) is the...Ch. 4.R - Suppose a particle moves back and forth along a...Ch. 4.R - a Explain the meaning of the indefinite integral...Ch. 4.R - Explain exactly what is meant by the statement...Ch. 4.R - State the Substitution Rule. In practice, how do...Ch. 4.R - Determine whether the statement is true or false....Ch. 4.R - Determine whether the statement is true or false....Ch. 4.R - Determine whether the statement is true or false....Ch. 4.R - Determine whether the statement is true or false....Ch. 4.R - Determine whether the statement is true or false....Ch. 4.R - Determine whether the statement is true or false....Ch. 4.R - Determine whether the statement is true or false....Ch. 4.R - Determine whether the statement is true or false....Ch. 4.R - Determine whether the statement is true or false....Ch. 4.R - Determine whether the statement is true or false....Ch. 4.R - Determine whether the statement is true or false....Ch. 4.R - Determine whether the statement is true or false....Ch. 4.R - Determine whether the statement is true or false....Ch. 4.R - Determine whether the statement is true or false....Ch. 4.R - Determine whether the statement is true or false....Ch. 4.R - Determine whether the statement is true or false....Ch. 4.R - Determine whether the statement is true or false....Ch. 4.R - Determine whether the statement is true or false....Ch. 4.R - Use the given graph of f to find the Riemann sum...Ch. 4.R - a Evaluate the Riemann sum for f(x)=x2x0x2 With...Ch. 4.R - Evaluate 01(x+1x2)dx By interpreting it in terms...Ch. 4.R - Express limni=1nsinxix as a definite integral on...Ch. 4.R - If 06f(x)dx=10 and 04f(x)dx=7, find 46f(x)dxCh. 4.R - a Write 15(x+2x5)dx as a limit of Riemann sums,...Ch. 4.R - The figure shows the graphs of f, f, and 0xf(t)dt....Ch. 4.R - Evaluate: a 0/2ddx(sinx2cosx3)dx b...Ch. 4.R - The graph of f consists of the three line segments...Ch. 4.R - If f is the function in Exercise 9, find g(4).Ch. 4.R - Evaluate the integral, if it exists. 12(8x3+3x2)dxCh. 4.R - Evaluate the integral, if it exists. 0T(x48x+7)dxCh. 4.R - Evaluate the integral, if it exists. 01(1x9)dxCh. 4.R - Evaluate the integral, if it exists. 01(1x)9dxCh. 4.R - Evaluate the integral, if it exists. 19u2u2uduCh. 4.R - Evaluate the integral, if it exists. 01(u4+1)2duCh. 4.R - Evaluate the integral, if it exists. 01y(y2+1)5dyCh. 4.R - Evaluate the integral, if it exists. 02y21+y3dyCh. 4.R - Evaluate the integral, if it exists. 15dt(t4)2Ch. 4.R - Evaluate the integral, if it exists. 01sin(3t)dtCh. 4.R - Evaluate the integral, if it exists. 01v2cos(v3)dvCh. 4.R - Evaluate the integral, if it exists. 11sinx1+x2dxCh. 4.R - Evaluate the integral, if it exists....Ch. 4.R - Evaluate the integral, if it exists. x+2x2+4xdxCh. 4.R - Evaluate the integral, if it exists. sintcostdtCh. 4.R - Evaluate the integral, if it exists....Ch. 4.R - Evaluate the integral, if it exists. 0/8sec2tan2dCh. 4.R - Evaluate the integral, if it exists....Ch. 4.R - Evaluate the integral, if it exists. 03|x24|dxCh. 4.R - Evaluate the integral, if it exists. 04|x1|dxCh. 4.R - Evaluate the indefinite integral. Illustrate and...Ch. 4.R - Evaluate the indefinite integral. Illustrate and...Ch. 4.R - Use a graph to give a rough estimate of the area...Ch. 4.R - Graph the function f(x)=cos2xsinx and use the...Ch. 4.R - Find the derivative of the function....Ch. 4.R - Find the derivative of the function....Ch. 4.R - Find the derivative of the function....Ch. 4.R - Find the derivative of the function....Ch. 4.R - Find the derivative of the function. y=xxcosdCh. 4.R - Find the derivative of the function....Ch. 4.R - Use property 8 of integrals to estimate the value...Ch. 4.R - Use property 8 of integrals to estimate the value...Ch. 4.R - Use the properties of integrals to verify the...Ch. 4.R - Use the properties of integrals to verify the...Ch. 4.R - Use the Midpoint Rule with n=6 to approximate...Ch. 4.R - A particle moves along a line with velocity...Ch. 4.R - Let r(t) be the rate at which the worlds oil is...Ch. 4.R - A radar gun was used to record the speed of a...Ch. 4.R - A population of honeybees increased at a rate of...Ch. 4.R - Let f(x)={x1if3x01x2if0x1 Evaluate 31f(x)dx by...Ch. 4.R - If f is continuous and 02f(x)dx=6, evaluate...Ch. 4.R - The Fresnel function S(x)=0xsin(12t2)dt was...Ch. 4.R - If f is a continuous function such that...Ch. 4.R - Find a function f and a value of the constant a...Ch. 4.R - If f is continuous on a, b, show that...Ch. 4.R - Find limh01h22+h1+t3dtCh. 4.R - If f is continuous on 0, 1, prove that...Ch. 4.R - Evaluate limn1n[(1n)9+(2n)9+(3n)9+...+(nn)9]Ch. 4.P - If xsinxx=0x2f(t)dt, where f is a continuous...Ch. 4.P - Find the minimum value of the area of the region...Ch. 4.P - If f is a differentiable function such that f(x)...Ch. 4.P - a Graph several members of the family of functions...Ch. 4.P - If f(x)=0g(x)11+t3dt, where...Ch. 4.P - If f(x)=0xx2sin(t2)dt, find f(x).Ch. 4.P - Find the interval a, b for which the value of the...Ch. 4.P - Use an integral to estimate the sum i=110000i.Ch. 4.P - a Evaluate 0nxdx, where n is a positive integer. b...Ch. 4.P - Find d2dx20x(1sint1+u4du)dt.Ch. 4.P - Suppose the coefficients of the cubic polynomial...Ch. 4.P - A circular disk of radius r is used in an...Ch. 4.P - Prove that if f is continuous, then...Ch. 4.P - The figure shows a parabolic segment, that is, a...Ch. 4.P - Given the point a, b in the first quadrant, find...Ch. 4.P - The figure shows a region consisting of all points...Ch. 4.P - Evaluate limn(1nn+1+1nn+2+...+1nn+n).Ch. 4.P - For any number c, we let fc(x) be the smaller of...

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