Chapter 5

In Problems 23-26, the velocity function for an object is given. Assuming that the object is at the origin at time \(t=0\), find the position at time \(t=4\). \(v(t)= \begin{cases}\sqrt{4-t^{2}} & \text { if } 0 \leq t \leq 2 \\ 0 & \text { if } 2

Prove the following formula for a geometric sum: $$ \sum_{k=0}^{n} a r^{k}=a+a r+a r^{2}+\cdots+a r^{n}=\frac{a-a r^{n+1}}{1-r}(r \neq 1) $$

In Problems 15-34, use the method of substitution to find each of the following indefinite integrals. \(\int x^{2} \cos \left(x^{3}+5\right) d x\)

Without doing any calculations, rank from smallest to largest the approximations of \(\int_{1}^{3}\left(x^{3}+x^{2}+x+1\right) d x\) for the following methods: left Riemann sum, right Riemann sum, Trapezoidal Rule, Parabolic Rule.

In Problems 17-26, find \(G^{\prime}(x)\). $$ G(x)=\int_{\cos x}^{\sin x} t^{5} d t $$

In Problems 15-34, use the method of substitution to find each of the following indefinite integrals. \(\int \frac{x \sin \sqrt{x^{2}+4}}{\sqrt{x^{2}+4}} d x\)

In Problems 27-32, find the interval(s) on which the graph of \(y=f(x), x \geq 0\), is (a) increasing, and (b) concave up. $$ f(x)=\int_{0}^{x} \frac{s}{\sqrt{1+s^{2}}} d s $$

In Problems 15-34, use the method of substitution to find each of the following indefinite integrals. \(\int \frac{z \cos \left(\sqrt[3]{z^{2}+3}\right)}{\left(\sqrt[3]{z^{2}+3}\right)^{2}} d z\)

In Problems 27-32, find the interval(s) on which the graph of \(y=f(x), x \geq 0\), is (a) increasing, and (b) concave up. $$ f(x)=\int_{0}^{x} \frac{1+t}{1+t^{2}} d t $$

In Problems 15-34, use the method of substitution to find each of the following indefinite integrals. \(\int x^{2}\left(x^{3}+5\right)^{8} \exp \left[\left(x^{3}+5\right)^{9}\right] d x\)

In Problems 27-32, find the interval(s) on which the graph of \(y=f(x), x \geq 0\), is (a) increasing, and (b) concave up. $$ f(x)=\int_{0}^{x} \tan ^{-1} u d u $$

In Problems 15-34, use the method of substitution to find each of the following indefinite integrals. \(\int x^{6}\left(7 x^{7}+\pi\right)^{8} \sinh \left[\left(7 x^{7}+\pi\right)^{9}\right] d x\)

\(\int_{0}^{2}\left[3+\sin \left(x^{2}\right)\right] d x\)

On her way to work, Teri noted her speed every 3 minutes. The results are shown in the table below. How far did she drive? \begin{tabular}{llllllllll} \hline Time (minutes) & 0 & 3 & 6 & 9 & 12 & 15 & 18 & 21 & 24 \\ Speed \((\mathrm{mi} / \mathrm{h})\) & 0 & 31 & 54 & 53 & 52 & 35 & 31 & 28 & 0 \\\ \hline \end{tabular}

Recall that \([x]\) denotes the greatest integer less than or equal to \(x\). Calculate each of the following integrals. You may use geometric reasoning and the fact that \(\int_{0}^{b} x^{2} d x=b^{3} / 3\). (The latter is shown in Problem 34.) (a) \(\int_{-3}^{3}[x] d x\) (b) \(\int_{-3}^{3}[x]^{2} d x\) (c) \(\int_{-3}^{3}(x-[x]) d x\) (d) \(\int_{-3}^{3}(x-[x])^{2} d x\) (e) \(\int_{-3}^{3}|x| d x\) (f) \(\int_{-3}^{3} x|x| d x\) (g) \(\int_{-1}^{2}|x|[x] d x\) (h) \(\int_{-1}^{2} x^{2}[x] d x\)

In Problems 15-34, use the method of substitution to find each of the following indefinite integrals. \(\int x \cos \left(x^{2}+4\right) \sqrt{\sin \left(x^{2}+4\right)} d x\)

\(\int_{-1}^{1} \frac{2}{1+x^{2}} d x\)

Every 12 minutes between 4:00 P.M. and 6:00 P.M., the rate (in gallons per minute) at which water flowed out of a town's water tank was measured. The results are shown in the table below. How much water was used in this 2-hour span? \begin{tabular}{lcccccc} \hline Time & \(4: 00\) & \(4: 12\) & \(4: 24\) & \(4: 36\) & \(4: 48\) & \(5: 00\) \\ Flow (gal/min) & 65 & 71 & 68 & 78 & 105 & 111 \\ \hline Time & \(5: 12\) & \(5: 24\) & \(5: 36\) & \(5: 48\) & \(6: 00\) & \\ Flow \((\mathrm{gal} / \mathrm{min})\) & 108 & 144 & 160 & 152 & 148 & \\ \hline \end{tabular}

In Problems 27-32, find the interval(s) on which the graph of \(y=f(x), x \geq 0\), is (a) increasing, and (b) concave up. $$ f(x)=\int_{1}^{x} \frac{1}{\theta} d \theta, x>0 $$

Let \(f\) be an odd function and \(g\) be an even function, and suppose that \(\int_{0}^{1}|f(x)| d x=\int_{0}^{1} g(x) d x=3\). Use geometric reasoning to calculate each of the following: (a) \(\int_{-1}^{1} f(x) d x\) (b) \(\int_{-1}^{1} g(x) d x\) (c) \(\int_{-1}^{1}|f(x)| d x\) (d) \(\int_{-1}^{1}[-g(x)] d x\) (e) \(\int_{-1}^{1} x g(x) d x\) (f) \(\int_{-1}^{1} f^{3}(x) g(x) d x\)

In Problems 15-34, use the method of substitution to find each of the following indefinite integrals. \(\int x^{6} \sin \left(3 x^{7}+9\right) \sqrt[3]{\cos \left(3 x^{7}+9\right)} d x\)

\(\int_{10}^{20}\left(1+\frac{1}{x}\right)^{5} d x\)

Show that \(\int_{a}^{b} x d x=\frac{1}{2}\left(b^{2}-a^{2}\right)\) by completing the following argument. For the partition \(a=x_{0}

In statistics we define the mean \(\bar{x}\) and the variance \(s^{2}\) of a sequence of numbers \(x_{1}, x_{2}, \ldots, x_{n}\) by $$ \bar{x}=\frac{1}{n} \sum_{i=1}^{n} x_{i}, \quad s^{2}=\frac{1}{n} \sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)^{2} $$ Find \(\bar{x}\) and \(s^{2}\) for the sequence of numbers \(2,5,7,8,9,10,14\).

In Problems 15-34, use the method of substitution to find each of the following indefinite integrals. \(\int x^{2} \sin \left(x^{3}+5\right) \cos ^{9}\left(x^{3}+5\right) d x\)

In Problems 15-34, use the method of substitution to find each of the following indefinite integrals. \(\int x^{-4} \sec ^{2}\left(x^{-3}+1\right) \sqrt[5]{\tan \left(x^{-3}+1\right)} d x\) Hint: \(D_{x} \tan x=\sec ^{2} x\)

In Problems 33-36, use the Interval Additive Property and linearity to evaluate \(\int_{0}^{4} f(x) d x\). Begin by drawing a graph of \(f\). $$ f(x)= \begin{cases}1 & \text { if } 0 \leq x<1 \\ x & \text { if } 1 \leq x<2 \\\ 4-x & \text { if } 2 \leq x \leq 4\end{cases} $$

CAS Many computer algebra systems permit the evaluation of Riemann sums for left end point, right end point, or midpoint evaluations of the function. Using such a system in Problems 35-38, evaluate the 10-subinterval Riemann sums using left end point, right end point, and midpoint evaluations. \(\int_{0}^{2}\left(x^{3}+1\right) d x\)

In Problems 35-62, use the Substitution Rule for Definite Integrals to evaluate each definite integral. \(\int_{0}^{1}\left(x^{2}+1\right)^{10}(2 x) d x \quad\)

In Problems \(35-44\), use symmetry to help you evaluate the given integral. 35\. \(\int_{-\pi}^{\pi}(\sin x+\cos x) d x\)

CAS Many computer algebra systems permit the evaluation of Riemann sums for left end point, right end point, or midpoint evaluations of the function. Using such a system in Problems 35-38, evaluate the 10-subinterval Riemann sums using left end point, right end point, and midpoint evaluations. \(\int_{0}^{1} \tan x d x\)

In Problems 35-62, use the Substitution Rule for Definite Integrals to evaluate each definite integral. \(\int_{-1}^{0} \sqrt{x^{3}+1}\left(3 x^{2}\right) d x\)

\(\int_{-1}^{1} \frac{x^{3}}{\left(1+x^{2}\right)^{4}} d x\)

In Problems 33-36, use the Interval Additive Property and linearity to evaluate \(\int_{0}^{4} f(x) d x\). Begin by drawing a graph of \(f\). $$ f(x)=3+|x-3| $$

Let \(x_{1}, x_{2}, \ldots, x_{n}\) be any real numbers. Find the value of \(c\) that minimizes \(\sum_{i=1}^{n}\left(x_{i}-c\right)^{2}\).

In Problems 35-62, use the Substitution Rule for Definite Integrals to evaluate each definite integral. \(\int_{-1}^{3} \frac{1}{(t+2)^{2}} d t\)

\(\int_{-\pi / 2}^{\pi / 2} \frac{\sin x}{1+\cos x} d x\)

CAS Many computer algebra systems permit the evaluation of Riemann sums for left end point, right end point, or midpoint evaluations of the function. Using such a system in Problems 35-38, evaluate the 10-subinterval Riemann sums using left end point, right end point, and midpoint evaluations. \(\int_{1}^{3}(1 / x) d x\)

In the song The Twelve Days of Christmas, my true love gave me 1 gift on the first day, \(1+2\) gifts on the second day, \(1+2+3\) gifts on the third day, and so on for 12 days. (a) Find the total number of gifts given in 12 days. (b) Find a simple formula for \(T_{n}\), the total number of gifts given during a Christmas of \(n\) days.

In Problems 35-62, use the Substitution Rule for Definite Integrals to evaluate each definite integral. \(\int_{2}^{10} \frac{1}{y+4} d y\)

\(\int_{-\sqrt[3]{\pi}}^{\sqrt[3]{\pi}} x^{2} \cos \left(x^{3}\right) d x\)

Prove that the function \(f\) defined by $$ f(x)= \begin{cases}1 & \text { if } x \text { is rational } \\ 0 & \text { if } x \text { is irrational }\end{cases} $$ is not integrable on \([0,1]\). Hint: Show that no matter how small the norm of the partition, \(\|P\|\), the Riemann sum can be made to have value either 0 or 1 .

A grocer stacks oranges in a pyramidlike pile. If the bottom layer is rectangular with 10 rows of 16 oranges and the top layer has a single row of oranges, how many oranges are in the stack?

In Problems 35-62, use the Substitution Rule for Definite Integrals to evaluate each definite integral. \(\int_{5}^{8} \sqrt{3 x+1} d x\)

$$ \int_{-\pi}^{\pi}(\sin x+\cos x)^{2} d x $$

Let \(F(x)=\int_{0}^{x}\left(t^{4}+1\right) d t\). (a) Find \(F(0)\). (b) Let \(y=F(x)\). Apply the First Fundamental Theorem of Calculus to obtain \(d y / d x=F^{\prime}(x)=x^{4}+1\). Solve the differential equation \(d y / d x=x^{4}+1\). (c) Find the solution to this differential equation that satisfies \(y=F(0)\) when \(x=0\). (d) Show that \(\int_{0}^{1}\left(x^{4}+1\right) d x=\frac{6}{5}\).

In Problems 35-62, use the Substitution Rule for Definite Integrals to evaluate each definite integral. \(\int_{1}^{7} \frac{1}{\sqrt{2 x+2}} d x\)

\(\int_{-\pi / 2}^{\pi / 2} z \sin ^{2}\left(z^{3}\right) \cos \left(z^{3}\right) d z\)

Let \(G(x)=\int_{0}^{x} \sin t d t\). (a) Find \(G(0)\) and \(G(2 \pi)\). (b) Let \(y=G(x)\). Apply the First Fundamental Theorem of Calculus to obtain \(d y / d x=G^{\prime}(x)=\sin x\). Solve the differential equation \(d y / d x=\sin x\). (c) Find the solution to this differential equation that satisfies \(y=G(0)\) when \(x=0\). (d) Show that \(\int_{0}^{\pi} \sin x d x=2\). (e) Find all relative extrema and inflection points of \(G\) on the interval \([0,4 \pi]\). (f) Plot a graph of \(y=G(x)\) over the interval \([0,4 \pi]\).

In Problems 35-62, use the Substitution Rule for Definite Integrals to evaluate each definite integral. \(\int_{-3}^{3} \sqrt{7+2 t^{2}}(8 t) d t\)