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CalculusSingle Variable Calculus(a) Write an expression for a Riemann sum of a function f on an interval [ a , b ]. Explain the meaning of the notation that you use. (b) If f ( x ) ⩾ 0 , what is the geometric interpretation of a Riemann sum? Illustrate with a diagram. (c) If f ( x ) takes on both positive and negative values, what is the geometric interpretation of a Riemann sum? Illustrate with a diagram.BuyFind*arrow_forward*

8th Edition

James Stewart

Publisher: Cengage Learning

ISBN: 9781305266636

Chapter 4, Problem 1RCC

Textbook Problem

(a) Write an expression for a Riemann sum of a function *f* on an interval [*a*, *b*]. Explain the meaning of the notation that you use.

(b) If

(c) If *f*(*x*) takes on both positive and negative values, what is the geometric interpretation of a Riemann sum? Illustrate with a diagram.

Expert Solution

(a)

To determine

**To find:** The expression for a Riemann sum of a function

The expression for a Riemann sum of a function *f* is

The Riemann sum of a function *f* is the method to find the total area underneath a curve.

The area under the curve dividedas *n* number of approximating rectangles. Hence the Riemann sum of a function *f* is the sum of the area of the all individual rectangles.

Here,

Thus, the expression for a Riemann sum of a function *f* is

Expert Solution

b)

To determine

**To define:** The geometric interpretation of a Riemann sum with diagram.

**Given information:**

Consider the condition for the function

**Explanation:**

The function

Sketch the curve *n* number approximating rectangles.

Show the curve as in Figure 1.

Refer to Figure 1

The function

Thus, the geometric interpretation of a Riemann sum of

Expert Solution

c)

To determine

**To define:** The geometric interpretation of a Riemann sum, if the function

**Given information:**

The function

**Explanation:**

The function

Sketch the curve *n* number approximating rectangles.

Show the curve as in Figure 2.

Refer figure 2,

The Riemann sum is the difference of areas of approximating rectangles above and below the *x-*axis

Therefore, the geometric interpretation of a Riemann sum is defined, if

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(x2+1+1x2+1)dxCh. 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. 7. x1x2dxCh. 4.5 - Evaluate the indefinite integral. 8. x2sin(x3)dxCh. 4.5 - Evaluate the indefinite integral. 9. 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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. 35. 01cos(t/2)dtCh. 4.5 - Evaluate the definite integral. 36. 01(3t1)50dtCh. 4.5 - Evaluate the definite integral. 37. 011+7x3dxCh. 4.5 - Evaluate the definite integral. 38. 0xcos(x2)dxCh. 4.5 - Evaluate the definite integral. 39. 0/6sintcos2tdtCh. 4.5 - Evaluate the definite integral. 40....Ch. 4.5 - Evaluate the definite integral. 41....Ch. 4.5 - Evaluate the definite integral. 42....Ch. 4.5 - Evaluate the definite integral. 43. 013dx(1+2x)23Ch. 4.5 - Evaluate the definite integral. 44. 0axa2x2dxCh. 4.5 - Evaluate the definite integral. 45. 0axx2+a2dx(a0)Ch. 4.5 - Evaluate the definite integral. 46. /3/3x4sinxdxCh. 4.5 - Evaluate the definite integral. 47. 12xx1dxCh. 4.5 - Evaluate the definite integral. 48. 04x1+2xdxCh. 4.5 - Evaluate the definite integral. 49....Ch. 4.5 - Evaluate the definite integral. 50....Ch. 4.5 - Evaluate the definite integral. 51. 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 on , prove that...Ch. 4.5 - If f is continuous 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 - Evaluate the integral. 67. dx53xCh. 4.5 - Evaluate the integral. 68. e5rdrCh. 4.5 - Evaluate the integral. 69. (lnx)2xdxCh. 4.5 - Evaluate the integral. 70. dxax+b(a0)Ch. 4.5 - Evaluate the integral. 71. ex1+exdxCh. 4.5 - Evaluate the integral. 72. ecostsintdtCh. 4.5 - Evaluate the integral. 73. (arctanx2)x2+1dxCh. 4.5 - Evaluate the integral. 74. xx2+4dxCh. 4.5 - Evaluate the integral. 75. 1+x1+x2dxCh. 4.5 - Evaluate the integral. 76. sin(lnx)xdxCh. 4.5 - Evaluate the integral. 77. sin2x1+cos2xdxCh. 4.5 - Evaluate the integral. 78. sinx1+cos2xdxCh. 4.5 - Evaluate the integral. 79. cotxdxCh. 4.5 - Evaluate the integral. 80. x1+x4dxCh. 4.5 - Evaluate the integral. 81. ee4dxxlnxCh. 4.5 - Evaluate the integral. 82. 01xex2dxCh. 4.5 - Evaluate the integral. 83. 01ez+1ez+zdzCh. 4.5 - Evaluate the integral. 84. 01(x1)e(x1)2dxCh. 4.5 - Use Exercise 64 to evaluate the integral...Ch. 4 - (a) Write an expression for a Riemann sum of a...Ch. 4 - (a) Write the definition of the definite integral...Ch. 4 - State the Midpoint Rule.Ch. 4 - State both parts of the Fundamental Theorem of...Ch. 4 - (a) State the Net Change Theorem. (b) If r(t) is...Ch. 4 - Suppose a particle moves back and forth along a...Ch. 4 - (a) Explain the meaning of the indefinite integral...Ch. 4 - Explain exactly what is meant by the statement...Ch. 4 - State the Substitution Rule. In practice, how do...Ch. 4 - Determine whether the statement is true or false....Ch. 4 - Determine whether the statement is true or false....Ch. 4 - Determine whether the statement is true or false....Ch. 4 - Determine whether the statement is true or false....Ch. 4 - Determine whether the statement is true or false....Ch. 4 - Determine whether the statement is true or false....Ch. 4 - Determine whether the statement is true or false....Ch. 4 - Determine whether the statement is true or false....Ch. 4 - Determine whether the statement is true or false....Ch. 4 - Determine whether the statement is true or false....Ch. 4 - Determine whether the statement is true or false....Ch. 4 - Determine whether the statement is true or false....Ch. 4 - Determine whether the statement is true or false....Ch. 4 - Determine whether the statement is true or false....Ch. 4 - Determine whether the statement is true or false....Ch. 4 - Determine whether the statement is true or false....Ch. 4 - Determine whether the statement is true or false....Ch. 4 - Determine whether the statement is true or false....Ch. 4 - Use the given graph of f to find the Riemann sum...Ch. 4 - (a) Evaluate the Riemann sum for f(x)=x2x0x2 with...Ch. 4 - Evaluate 01(x+1x2)dx by interpreting it in terms...Ch. 4 - Express limni=1nsinxix as a definite integral on...Ch. 4 - If 06f(x)dx=10 and04f(x)dx=7, find46f(x)dx.Ch. 4 - (a) Write 15(x+2x5)dx as a limit of Riemann sums,...Ch. 4 - The figure shows the graphs of f,f, and 0xf(t)dt....Ch. 4 - Evaluate: (a) 0/2ddx(sinx2cosx3)dx (b)...Ch. 4 - The graph of f consists of the three line segments...Ch. 4 - If f is the function in Exercise 9, find g(4).Ch. 4 - Evaluate the integral, if it exists. 11....Ch. 4 - Evaluate the integral, if it exists. 12....Ch. 4 - Evaluate the integral, if it exists. 13. 01(1x9)dxCh. 4 - Evaluate the integral, if it exists. 14. 01(1x)9dxCh. 4 - Evaluate the integral, if it exists. 15. 19u2u2uduCh. 4 - Evaluate the integral, if it exists. 16....Ch. 4 - Evaluate the integral, if it exists. 17....Ch. 4 - Evaluate the integral, if it exists. 18....Ch. 4 - Evaluate the integral, if it exists. 19. 15dt(t4)2Ch. 4 - Evaluate the integral, if it exists. 20....Ch. 4 - Evaluate the integral, if it exists. 21....Ch. 4 - Evaluate the integral, if it exists. 22....Ch. 4 - Evaluate the integral, if it exists. 23....Ch. 4 - Evaluate the integral, if it exists. 24....Ch. 4 - Evaluate the integral, if it exists. 25....Ch. 4 - Evaluate the integral, if it exists. 26....Ch. 4 - Evaluate the integral, if it exists. 27....Ch. 4 - Evaluate the integral, if it exists. 28....Ch. 4 - Evaluate the integral, if it exists. 29. 03|x24|dxCh. 4 - Evaluate the integral, if it exists. 30. 04|x1|dxCh. 4 - Evaluate the indefinite integral. Illustrate and...Ch. 4 - Evaluate the indefinite integral. Illustrate and...Ch. 4 - Use a graph to give a rough estimate of the area...Ch. 4 - Graph the function f(x) = cos2x sin x and use the...Ch. 4 - Find the derivative of the function. 35....Ch. 4 - Find the derivative of the function. 36....Ch. 4 - Find the derivative of the function. 37....Ch. 4 - Find the derivative of the function. 38....Ch. 4 - Find the derivative of the function. 39.y=xxcosdCh. 4 - Find the derivative of the function. 40....Ch. 4 - Use Property 8 of integrals to estimate the value...Ch. 4 - Use Property 8 of integrals to estimate the value...Ch. 4 - Use the properties of integrals to verify the...Ch. 4 - Use the properties of integrals to verify the...Ch. 4 - Use the Midpoint Rule with n = 6 to approximate...Ch. 4 - A particle moves along a line with velocity...Ch. 4 - Let r(t) be the rate at which the worlds oil is...Ch. 4 - A radar gun was used to record the speed of a...Ch. 4 - A population of honeybees increased at a rate of...Ch. 4 - Let f(x)={x1if3x01x2if0x1 Evaluate 31f(x)dx by...Ch. 4 - If f is continuous and 02f(x)dx=6, evaluate...Ch. 4 - The Fresnel function S(x)=0xsin(12t2)dt was...Ch. 4 - If f is a continuous function such that...Ch. 4 - Find a function f and a value of the constant a...Ch. 4 - If f' is continuous on [a, b], show that...Ch. 4 - Find limh01h22+h1+t3dtCh. 4 - If f is continuous on [0, 1], prove that...Ch. 4 - Evaluate limn1n[(1n)9+(2n)9+(3n)9+...+(nn)9]Ch. 4 - If xsinx=0x2f(t)dt, where f is a continuous...Ch. 4 - Find the minimum value of the area of the region...Ch. 4 - If f is a differentiable function such that f(x)...Ch. 4 - (a) Graph several members of the family of...Ch. 4 - If f(x)0g(x)11+t3dt, where...Ch. 4 - If f(x)=0xx2sin(t2)dt, find f(x).Ch. 4 - Find the interval [a, b] for which the value of...Ch. 4 - Use an integrai to estimate th sum i=110000i.Ch. 4 - (a) Evaluate 0nxdx, where n is a positive integer....Ch. 4 - Find d2dx20x(1sint1+u4du)dt.Ch. 4 - Suppose the coefficients of the cubic polynomial...Ch. 4 - A circular disk of radius r is used in an...Ch. 4 - Prove that if f is continuous, then...Ch. 4 - The figure shows a parabolic segment, that is. a...Ch. 4 - Given the point (a, b) in the first quadrant, find...Ch. 4 - The figure shows a region consisting of all points...Ch. 4 - Evaluate limn(1nn+1+1nn+2++1nn+n).Ch. 4 - For any number c, we let fc(x) be the smaller of...

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