Problem 1QC: In Example 1, let B(x) be the area function for f with left endpoint 5. Evaluate B(5) and B(9).... Problem 2QC: Verify that the area function in Example 2c gives the correct area when x = 6 and x = 10. Example 2... Problem 3QC: Evaluate (xx+1)|12. Problem 4QC: Explain why f is an antiderivative of f. Problem 1E: Suppose A is an area function of f. What is the relationship between f and A? Problem 2E: Suppose F is an antiderivative of f and A is an area function of f. What is the relationship between... Problem 3E: Explain in words and write mathematically how the Fundamental Theorem of Calculus is used to... Problem 4E: Let f(x) = c, where c is a positive constant. Explain why an area function of f is an increasing... Problem 5E: The linear function f(x) = 3 x is decreasing on the interval [0, 3]. Is the area function for f... Problem 6E: Evaluate 023x2dx and 223x2dx. Problem 7E: Explain in words and express mathematically the inverse relationship between differentiation and... Problem 8E: Why can the constant of integration be omitted from the antiderivative when evaluating a definite... Problem 9E: Evaluate ddxaxf(t)dt and ddxabf(t)dt, where a and b are constants. Problem 10E: Explain why abf(x)dx=f(b)f(a). Problem 11E: Evaluate 38f(t)dt, where f is continuous on [3, 8], f(3) = 4, and f(8) = 20. Problem 12E: Evaluate 273dx using the Fundamental Theorem of Calculus. Check your work by evaluating the integral... Problem 13E Problem 14E: Area functions The graph of f is shown in the figure. Let A(x)=0xf(t)dt and F(x)=2xf(t)dt be two... Problem 15E: Area functions for constant functions Consider the following functions f and real numbers a (see... Problem 16E: Area functions for constant functions Consider the following functions f and real numbers a (see... Problem 17E: Area functions for the same linear function Let f(t) = t and consider the two area functions... Problem 18E: Area functions for the same linear function Let f(t) = 2t 2 and consider the two area functions... Problem 19E: Area functions for linear functions Consider the following functions f and real numbers a (see... Problem 20E: Area functions for linear functions Consider the following functions f and real numbers a (see... Problem 21E: Area functions for linear functions Consider the following functions f and real numbers a (see... Problem 22E: Area functions for linear functions Consider the following functions f and real numbers a (see... Problem 23E: Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus.... Problem 24E: Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus.... Problem 25E: Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus.... Problem 26E: Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus.... Problem 27E: Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus.... Problem 28E: Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus.... Problem 29E: Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus. 29.... Problem 30E: Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus. 30.... Problem 31E: Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus.... Problem 32E: Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus.... Problem 33E: Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus. 31.... Problem 34E: Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus. 32.... Problem 35E: Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus. 33.... Problem 36E: Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus. 34.... Problem 37E: Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus. 35.... Problem 38E: Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus. 36.... Problem 39E: Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus.... Problem 40E: Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus.... Problem 41E: Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus. 43.... Problem 42E: Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus. 40.... Problem 43E: Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus. 41.... Problem 44E: Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus. 42.... Problem 45E: Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus. 39.... Problem 46E: Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus. 44.... Problem 47E: Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus. 45.... Problem 48E: Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus.... Problem 49E: Definite integrals Evaluate the following definite integrals using the Fundamental Theorem of... Problem 50E: Definite integrals Evaluate the following definite integrals using the Fundamental Theorem of... Problem 51E: Definite integrals Evaluate the following definite integrals using the Fundamental Theorem of... Problem 52E: Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus.... Problem 53E: Definite integrals Evaluate the following definite integrals using the Fundamental Theorem of... Problem 54E: Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus.... Problem 55E: Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus.... Problem 56E: Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus.... Problem 57E: Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus.... Problem 58E: Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus.... Problem 59E: Definite integrals Evaluate the following definite integrals using the Fundamental Theorem of... Problem 60E: Definite integrals Evaluate the following definite integrals using the Fundamental Theorem of... Problem 61E: Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus.... Problem 62E: Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus.... Problem 63E: Areas Find (i) the net area and (ii) the area of the following regions. Graph the function and... Problem 64E: Areas Find (i) the net area and (ii) the area of the following regions. Graph the function and... Problem 65E: Areas Find (i) the net area and (ii) the area of the following regions. Graph the function and... Problem 66E: Areas Find (i) the net area and (ii) the area of the following regions. Graph the function and... Problem 67E: Areas of regions Find the area of the region bounded by the graph of f and the x-axis on the given... Problem 68E: Areas of regions Find the area of the region bounded by the graph of f and the x-axis on the given... Problem 69E: Areas of regions Find the area of the region bounded by the graph of f and the x-axis on the given... Problem 70E: Areas of regions Find the area of the region bounded by the graph of f and the x-axis on the given... Problem 71E: Areas of regions Find the area of the region bounded by the graph of f and the x-axis on the given... Problem 72E: Areas of regions Find the area of the region bounded by the graph of f and the x-axis on the given... Problem 73E: Derivatives of integrals Simplify the following expressions. 61. ddx3x(t2+t+1)dt Problem 74E: Derivatives and integrals Simplify the given expressions. 102. ddxx1et2dt Problem 75E: Derivatives of integrals Simplify the following expressions. 65. ddxx1t4+1dt Problem 76E: Derivatives of integrals Simplify the following expressions. 66. ddxx0dpp2+1 Problem 77E: Derivatives of integrals Simplify the following expressions. 63. ddx2x3dpp2 Problem 78E: Derivatives and integrals Simplify the given expressions. 100. ddx0x2dtt2+4 Problem 79E: Derivatives and integrals Simplify the given expressions. 101. ddx0cosx(t4+6)dt Problem 80E: Derivatives of integrals Simplify the following expressions. 80.ddw0wln(x2+1)dx Problem 81E: Derivatives of integrals Simplify the following expressions. 81.ddzsinz10dtt4+1 Problem 82E: Derivatives of integrals Simplify the following expressions. 82.ddyy310x6+1dx Problem 83E: Derivatives and integrals Simplify the given expressions. 103. ddt(1t3xdxt213xdx) Problem 84E: Derivatives and integrals Simplify the given expressions. 104. ddt(0tdx1+x2+01/tdx1+x2) Problem 85E: Derivatives of integrals Simplify the following expressions. 85.ddx0x1+t2dt Problem 86E: Derivatives of integrals Simplify the following expressions. 68. ddxete2nlnt2dt Problem 87E Problem 88E: Working with area functions Consider the function f and its graph. a. Estimate the zeros of the area... Problem 89E Problem 90E Problem 91E Problem 92E Problem 93E: Area functions from graphs The graph of f is given in the figure. Let A(x)=0xf(t)dt and evaluate... Problem 94E Problem 95E: Working with area functions Consider the function f and the points a, b, and c. a. Find the area... Problem 96E: Working with area functions Consider the function f and the points a, b, and c. a. Find the area... Problem 97E Problem 98E Problem 99E: Find the critical points of the function f(x)=1xt2(t3)(t4)dt, and determine the intervals on which f... Problem 100E: Determine the intervals on which the function g(x)=x0tt2+1dt is concave up or concave down. Problem 101E Problem 102E Problem 103E: Areas of regions Find the area of the region R bounded by the graph of f and the x-axis on the given... Problem 104E Problem 105E: Areas of regions Find the area of the region R bounded by the graph of f and the x-axis on the given... Problem 106E: Areas of regions Find the area of the region R bounded by the graph of f and the x-axis on the given... Problem 107E: Explain why or why not Determine whether the following statements are true and give an explanation... Problem 108E: Explorations and Challenges Evaluate limx22xt2+t+3dtx24. Problem 109E: Maximum net area What value of b 1 maximizes the integral 1bx2(3x)dx? Problem 110E: Maximum net area Graph the function f(x) = 8 + 2x x2 and determine the values of a and b that... Problem 111E Problem 112E: Cubic zero net area Consider the graph of the cubic y = x(x a)(x b), where 0 a b. Verify that... Problem 113E: An integral equation Use the Fundamental Theorem of Calculus, Part 1, to find the function f that... Problem 114E Problem 115E: Asymptote of sine integral Use a calculator to approximate limxS(x)=limx0xsinttdt, where S is the... Problem 116E: Sine integral Show that the sine integral S(x)=0xsinttdt satisfies the (differential) equation xS(x)... Problem 117E Problem 118E: Continuity at the endpoints Assume that f is continuous on [a, b] and let A be the area function for... Problem 119E: Discrete version of the Fundamental Theorem In this exercise, we work with a discrete problem and... format_list_bulleted