Chapter 5

Use a change of variables to evaluate the following integrals. $$\int_{2}^{3} \frac{x}{\sqrt[3]{x^{2}-1}} d x$$

Displacement from velocity The following functions describe the velocity of a car (in mi/ hr) moving along a straight highway for a 3 -hr interval. In each case, find the function that gives the displacement of the car over the interval \([0, t],\) where \(0 \leq t \leq 3\). $$v(t)=\left\\{\begin{array}{ll}30 & \text { if } 0 \leq t \leq 2 \\\50 & \text { if } 2 < t \leq 2.5 \\\44 & \text { if } 2.5 < t \leq 3\end{array}\right.$$.

Use a change of variables to evaluate the following integrals. $$\int_{0}^{6 / 5} \frac{d x}{25 x^{2}+36}$$

Use geometry to evaluate the following integrals. $$\int_{-2}^{3}|x+1| d x$$

Use a change of variables to evaluate the following integrals. $$\int_{-\pi}^{0} \frac{\sin x}{2+\cos x} d x$$

Functions with absolute value Use a calculator and the method of your choice to approximate the area of the following regions. Present your calculations in a table, showing approximations using \(n=16,32,\) and 64 subintervals. Comment on whether your approximations appear to approach a limit.The region bounded by the graph of \(f(x)=\left|x\left(x^{2}-1\right)\right|\) and the \(x\) -axis on the interval \([-1,1]\).

Use geometry to evaluate the following integrals. $$\int_{1}^{6}|2 x-4| d x$$

Use a change of variables to evaluate the following integrals. $$\int_{-1}^{1}(x-1)\left(x^{2}-2 x\right)^{7} d x$$

Functions with absolute value Use a calculator and the method of your choice to approximate the area of the following regions. Present your calculations in a table, showing approximations using \(n=16,32,\) and 64 subintervals. Comment on whether your approximations appear to approach a limit.The region bounded by the graph of \(f(x)=|\cos 2 x|\) and the \(x\) -axis on the interval \([0, \pi]\).

Use geometry to evaluate the following integrals. $$\int_{1}^{6}(3 x-6) d x$$

Use a change of variables to evaluate the following integrals. $$\int_{-\pi}^{0} \frac{\sin x}{2+\cos x} d x$$

Functions with absolute value Use a calculator and the method of your choice to approximate the area of the following regions. Present your calculations in a table, showing approximations using \(n=16,32,\) and 64 subintervals. Comment on whether your approximations appear to approach a limit.The region bounded by the graph of \(f(x)=\left|1-x^{3}\right|\) and the \(x\) -axis on the interval \([-1,2]\).

Use geometry to evaluate the following integrals. $$\int_{-6}^{4} \sqrt{24-2 x-x^{2}} d x$$

Use a change of variables to evaluate the following integrals. $$\int_{0}^{1} \frac{(v+1)(v+2)}{2 v^{3}+9 v^{2}+12 v+36} d v$$

Let \(f(x)=c,\) where \(c>0,\) be a constant function on \([a, b] .\) Prove that any Riemann sum for any value of \(n\) gives the exact area of the region between the graph of \(f\) and the \(x\) -axis on \([a, b]\).

Suppose \(f\) is continuous on the intervals \([a, p]\) and \((p, b],\) where \(a < p < b\) with a finite jump at \(p .\) Form a uniform partition on the interval \([a, p]\) with \(n\) grid points and another uniform partition on the interval \([p, b]\) with \(m\) grid points, where \(p\) is a grid point of both partitions. Write a Riemann sum for \(\int_{a}^{b} f(x) d x\) and separate it into two pieces for \([a, p]\) and \([p, b] .\) Explain why \(\int_{a}^{b} f(x) d x=\int_{a}^{p} f(x) d x+\int_{p}^{b} f(x) d x\)

Use a change of variables to evaluate the following integrals. $$\int_{1}^{2} \frac{4}{9 x^{2}+6 x+1} d x$$

Assume that the linear function \(f(x)=m x+c\) is positive on the interval \([a, b] .\) Prove that the midpoint Riemann sum with any value of \(n\) gives the exact area of the region between the graph of \(f\) and the \(x\) -axis on \([a, b]\).

Use geometry and the result of Exercise 76 to evaluate the following integrals. $$\int_{0}^{10} f(x) d x, \text { where } f(x)=\left\\{\begin{array}{ll} 2 & \text { if } 0 \leq x \leq 5 \\ 3 & \text { if } 5 < x \leq 10 \end{array}\right.$$

Use a change of variables to evaluate the following integrals. $$\int_{0}^{\pi / 4} e^{\sin ^{2} x} \sin 2 x d x$$

Consider the function \(f\) and the points \(a, b,\) and \(c\) a. Find the area function \(A(x)=\int_{a}^{x} f(t) d t\) using the Fundamental Theorem. b. Graph \(f\) and \(A\) c. Evaluate \(A(b)\) and \(A(c)\). Interpret the results using the graphs of part\((b)\) $$f(x)=-12 x(x-1)(x-2) ; a=0, b=1, c=2$$

Use geometry and the result of Exercise 76 to evaluate the following integrals. $$\int_{1}^{6} f(x) d x, \text { where } f(x)=\left\\{\begin{array}{ll} 2 x & \text { if } 1 \leq x<4 \\ 10-2 x & \text { if } 4 \leq x \leq 6 \end{array}\right.$$

Find the area of the following regions. The region bounded by the graph of \(f(x)=x \sin x^{2}\) and the \(x\) -axis between \(x=0\) and \(x=\sqrt{\pi}\)

Consider the function \(f\) and the points \(a, b,\) and \(c\) a. Find the area function \(A(x)=\int_{a}^{x} f(t) d t\) using the Fundamental Theorem. b. Graph \(f\) and \(A\) c. Evaluate \(A(b)\) and \(A(c)\). Interpret the results using the graphs of part\((b)\) $$f(x)=\cos \pi x ; a=0, b=\frac{1}{2}, c=1$$

Recall that the floor function \(\lfloor x\rfloor\) is the greatest integer less than or equal to \(x\) and that the ceiling function \(\lceil x\rceil\) is the least integer greater than or equal to \(x\). Use the result of Exercise 76 and the graphs to evaluate the following integrals. $$\int_{1}^{5} x\lfloor x\rfloor d x$$

Find the area of the following regions. The region bounded by the graph of \(f(\theta)=\cos \theta \sin \theta\) and the \(\theta\) -axis between \(\theta=0\) and \(\theta=\pi / 2\)

Consider the function \(f\) and the points \(a, b,\) and \(c\) a. Find the area function \(A(x)=\int_{a}^{x} f(t) d t\) using the Fundamental Theorem. b. Graph \(f\) and \(A\) c. Evaluate \(A(b)\) and \(A(c)\). Interpret the results using the graphs of part\((b)\) $$f(x)=\frac{1}{x} ; a=1, b=4, c=6$$

Recall that the floor function \(\lfloor x\rfloor\) is the greatest integer less than or equal to \(x\) and that the ceiling function \(\lceil x\rceil\) is the least integer greater than or equal to \(x\). Use the result of Exercise 76 and the graphs to evaluate the following integrals. $$\int_{0}^{4} \frac{x}{\lceil x\rceil} d x$$

Find the area of the following regions. The region bounded by the graph of \(f(x)=(x-4)^{4}\) and the \(x\) -axis between \(x=2\) and \(x=6\)

Consider the function g, which is given in terms of a definite integral with a variable upper limit. a. Graph the integrand. b. Calculate \(g^{\prime}(x)\) c. Graph g, showing all your work and reasoning. $$g(x)=\int_{0}^{x} \sin ^{2} t d t$$

Use the definition of the definite integral to justify the property \(\int_{a}^{b} c f(x) d x=c \int_{a}^{b} f(x) d x,\) where \(f\) is continuous and \(c\) is a real number.

Find the area of the following regions. The region bounded by the graph of \(f(x)=\frac{x}{\sqrt{x^{2}-9}}\) and the \(x\) -axis between \(x=4\) and \(x=5\)

Consider the function g, which is given in terms of a definite integral with a variable upper limit. a. Graph the integrand. b. Calculate \(g^{\prime}(x)\) c. Graph g, showing all your work and reasoning. $$g(x)=\int_{0}^{x}\left(t^{2}+1\right) d t$$

Morphing parabolas The family of parabolas \(y=(1 / a)-x^{2} / a^{3}\) where \(a>0,\) has the property that for \(x \geq 0,\) the \(x\) -intercept is \((a, 0)\) and the \(y\) -intercept is \((0,1 / a) .\) Let \(A(a)\) be the area of the region in the first quadrant bounded by the parabola and the \(x\) -axis. Find \(A(a)\) and determine whether it is an increasing, decreasing, or constant function of \(a\)

Consider the function defined on [0,1] such that \(f(x)=1\) if \(x\) is a rational number and \(f(x)=0\) if \(x\) is irrational. This function has an infinite number of discontinuities, and the integral \(\int_{0}^{1} f(x) d x\) does not exist. Show that the right, left, and midpoint Riemann sums on regular partitions with \(n\) subintervals equal 1 for all \(n\). (Hint: Between any two real numbers lie a rational and an irrational number.)

Substitutions Suppose that \(f\) is an even function with \(\int_{0}^{8} f(x) d x=9 .\) Evaluate each integral. a. \(\int_{-1}^{1} x f\left(x^{2}\right) d x\) b. \(\int_{-2}^{2} x^{2} f\left(x^{3}\right) d x\)

Consider the function g, which is given in terms of a definite integral with a variable upper limit. a. Graph the integrand. b. Calculate \(g^{\prime}(x)\) c. Graph g, showing all your work and reasoning. $$g(x)=\int_{0}^{x} \cos (\pi \sqrt{t}) d t$$

Consider the integral \(I(p)=\int_{0}^{1} x^{p} d x,\) where \(p\) is a positive integer. a. Write the left Riemann sum for the integral with \(n\) subintervals. b. It is a fact (proved by the 17 th-century mathematicians Fermat and Pascal) that $$\lim _{n \rightarrow \infty} \frac{1}{n} \sum_{k=0}^{n-1}\left(\frac{k}{n}\right)^{p}=\frac{1}{p+1} .$$ Use this fact to evaluate \(I(p) . \quad \quad \quad\)

Substitutions Suppose that \(p\) is a nonzero real number and \(f\) is an odd integrable function with \(\int_{0}^{1} f(x) d x=\pi .\) Evaluate each integral. a. \(\int_{0}^{\pi /(2 p)} \cos p x f(\sin p x) d x\) b. \(\int_{-\pi / 2}^{\pi / 2} \cos x f(\sin x) d x\)

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. Suppose that \(f\) is a positive decreasing function, for \(x>0\) Then the area function \(A(x)=\int_{0}^{x} f(t) d t\) is an increasing function of \(x\) b. Suppose that \(f\) is a negative increasing function, for \(x>0\) Then the area function \(A(x)=\int_{0}^{x} f(t) d t\) is a decreasing function of \(x\) c. The functions \(p(x)=\sin 3 x\) and \(q(x)=4 \sin 3 x\) are antiderivatives of the same function. d. If \(A(x)=3 x^{2}-x-3\) is an area function for \(f,\) then \(B(x)=3 x^{2}-x\) is also an area function for \(f\) e. \(\frac{d}{d x} \int_{a}^{b} f(t) d t=0\)

Evaluate \(\int_{a}^{b} \frac{d x}{x^{2}},\) where \(0 < a < b,\) using the definition of the definite integral and the following steps. a. Assume \(\left\\{x_{0}, x_{1}, \ldots, x_{n}\right\\}\) is a partition of \([a, b]\) with \(\Delta x_{k}=x_{k}-x_{k-1},\) for \(k=1,2, \ldots, n .\) Show that \(x_{k-1} \leq \sqrt{x_{k-1} x_{k}} \leq x_{k},\) for \(k=1,2, \ldots, n\) b. Show that \(\frac{1}{x_{k-1}}-\frac{1}{x_{k}}=\frac{\Delta x_{k}}{x_{k-1} x_{k}},\) for \(k=1,2, \ldots, n\) c. Simplify the general Riemann sum for \(\int_{a}^{b} \frac{d x}{x^{2}}\) using \(\bar{x}_{k}^{*}=\sqrt{x_{k-1} x_{k}}\) d. Conclude that \(\int_{a}^{b} \frac{d x}{x^{2}}=\frac{1}{a}-\frac{1}{b}\)

Periodic motion An object moves along a line with a velocity in \(\mathrm{m} / \mathrm{s}\) given by \(v(t)=8 \cos (\pi t / 6) .\) Its initial position is \(s(0)=0\) a. Graph the velocity function. b. As discussed in Chapter \(6,\) the position of the object is given by \(s(t)=\int_{0}^{t} v(y) d y,\) for \(t \geq 0 .\) Find the position function, for \(t \geq 0\) c. What is the period of the motion - that is, starting at any point, how long does it take the object to return to that position?

Evaluate the following definite integrals using the Fundamental Theorem of Calculus. $$\frac{1}{2} \int_{0}^{\ln 2} e^{x} d x$$

Evaluate the following definite integrals using the Fundamental Theorem of Calculus. $$\int_{1}^{4} \frac{x-2}{\sqrt{x}} d x$$

Average distance on a triangle Consider the right triangle with vertices \((0,0),(0, b),\) and \((a, 0),\) where \(a>0\) and \(b>0 .\) Show that the average vertical distance from points on the \(x\) -axis to the hypotenuse is \(b / 2,\) for all \(a>0\)

Evaluate the following definite integrals using the Fundamental Theorem of Calculus. $$\int_{1}^{2}\left(\frac{2}{s}-\frac{4}{s^{3}}\right) d s$$

Average value of sine functions Use a graphing utility to verify that the functions \(f(x)=\sin k x\) have a period of \(2 \pi / k,\) where \(k=1,2,3, \ldots . .\) Equivalently, the first "hump" of \(f(x)=\sin k x\) occurs on the interval \([0, \pi / k] .\) Verify that the average value of the first hump of \(f(x)=\sin k x\) is independent of \(k .\) What is the average value?

Evaluate the following definite integrals using the Fundamental Theorem of Calculus. $$\int_{0}^{\pi / 3} \sec x \tan x d x$$

Looking ahead: Integrals of \(\tan x\) and cot \(x\) Use a change of variables to verify each integral. a. \(\int \tan x d x=-\ln |\cos x|+C=\ln |\sec x|+C\) b. \(\int \cot x d x=\ln |\sin x|+C\)

Evaluate the following definite integrals using the Fundamental Theorem of Calculus. $$\int_{\pi / 4}^{\pi / 2} \csc ^{2} \theta d \theta$$