Chapter 5

Calculate \(\int_{a}^{b} f(x) d x\), where a and \(b\) are the left and right end points for which f is defined, by using the Interval Additive Property and the appropriate area formulas from plane geometry. Begin by graphing the given function. $$ f(x)=\sqrt{A^{2}-x^{2}} ;-A \leq x \leq A $$

Find \(G^{\prime}(x)\). $$ G(x)=\int_{1}^{x} x t d t \text { (Be careful.) } $$

Find all values of \(c\) that satisfy the Mean Value Theorem for Integrals on the given interval. $$ g(y)=\cos 2 y ; \quad[0, \pi] $$

use the method of substitution to find each of the following indefinite integrals. $$ \int x^{2}\left(x^{3}+5\right)^{9} d x $$

Calculate \(\int_{a}^{b} f(x) d x\), where a and \(b\) are the left and right end points for which f is defined, by using the Interval Additive Property and the appropriate area formulas from plane geometry. Begin by graphing the given function. $$ f(x)=4-|x|,-4 \leq x \leq 4 $$

Find \(G^{\prime}(x)\). $$ G(x)=\int_{1}^{x^{2}} e^{-t^{2}} d t $$

Find all values of \(c\) that satisfy the Mean Value Theorem for Integrals on the given interval. $$ R(v)=v^{2}-v ; \quad[0,2] $$

use the method of substitution to find each of the following indefinite integrals. $$ \int x\left(x^{2}+3\right)^{-12 / 7} d x $$

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 \(t=4\). $$ v(t)=t / 60 $$

Find all values of \(c\) that satisfy the Mean Value Theorem for Integrals on the given interval. $$ T(x)=x^{3} ; \quad[0,2] $$

Find \(G^{\prime}(x)\). $$ G(x)=\int_{1}^{x^{2}+x} \sqrt{2 z+\sin z} d z $$

use the method of substitution to find each of the following indefinite integrals. $$ \int v\left(\sqrt{3} v^{2}+\pi\right)^{7 / 8} d v $$

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 \(t=4\). $$ v(t)=1+2 t $$

Add both sides of the two equalities below, solve for \(S\). and thereby give another proof of Formula \(1 .\) $$ \begin{array}{l} S=1+2+3+\cdots+(n-2)+(n-1)+n \\ S=n+(n-1)+(n-2)+\cdots+3+2+1 \end{array} $$

Without doing any calculations, rank from smallest to largest the approximations of \(\int_{0}^{1} \sqrt{x^{2}+1} d x\) for the following methods: left Riemann sum, right Riemann sum, midpoint Riemann sum, Trapezoidal Rule.

Find \(G^{\prime}(x)\). $$ G(x)=\int_{-x^{2}}^{x} \frac{t^{2}}{1+t^{2}} d t $$

Find all values of \(c\) that satisfy the Mean Value Theorem for Integrals on the given interval. $$ f(x)=a x+b ; \quad[1,4] $$

use the method of substitution to find each of the following indefinite integrals. $$ \int x \sin \left(x^{2}+4\right) d x $$

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)=\left\\{\begin{array}{ll} t / 2 & \text { if } 0 \leq t \leq 2 \\ 1 & \text { if } 2

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

Find \(G^{\prime}(x)\). $$ G(x)=\int_{\cos x}^{\sin x} t^{5} d t $$

Find all values of \(c\) that satisfy the Mean Value Theorem for Integrals on the given interval. $$ S(y)=y^{2} ; \quad[0, b] $$

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

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)=\left\\{\begin{array}{ll} \sqrt{4-t^{2}} & \text { if } 0 \leq t \leq 2 \\ 0 & \text { if } 2

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

use the method of substitution to find each of the following indefinite integrals. $$ \int \frac{x \sin \sqrt{x^{2}+4}}{\sqrt{x^{2}+4}} d x $$

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

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

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

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

Find the interval(s) on which the graph of \(y=f(x), x \geq 0\), is (a) increasing, and (b) concave up. $$ f(x)=\int_{0}^{x}(t+\sin t) d t $$

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

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

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

Recall that \([x \rrbracket\) 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) \(\int_{-3}^{3} \llbracket x \rrbracket d x\) (b) \(\int_{-3}^{3}[x]^{2} d x\) (c) \(\int_{-3}^{3}(x-[x \rrbracket) d x\) (d) \(\int_{-3}^{3}(x-\llbracket x \rrbracket)^{2} d x\) (e) \(\int_{-3}^{3}|x| d x\) (f) \(\int_{-3}^{3} x|x| d x\) (g) \(\int_{-1}^{2}|x| \llbracket x \rrbracket d x\) (h) \(\int_{-1}^{2} x^{2}[x \rrbracket d x\)

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

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 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 \(2,5,7,8,9,10,14\).

Use the Interval Additive Property and linearity to evaluate \(\int_{0}^{4} f(x) d x .\) Begin by drawing a graph of \(f\). $$ f(x)=\left\\{\begin{array}{ll} 2 & \text { if } 0 \leq x<2 \\ x & \text { if } 2 \leq x \leq 4 \end{array}\right. $$

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

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

Use the Interval Additive Property and linearity to evaluate \(\int_{0}^{4} f(x) d x .\) Begin by drawing a graph of \(f\). $$ f(x)=\left\\{\begin{array}{ll} 1 & \text { if } 0 \leq x<1 \\ x & \text { if } 1 \leq x<2 \\ 4-x & \text { if } 2 \leq x \leq 4 \end{array}\right. $$

use the method of substitution to find each of the following indefinite integrals. $$ \int x^{-4} \sec ^{2}\left(x^{-3}+1\right) \sqrt[5]{\tan \left(x^{-3}+1\right)} d x $$

Use symmetry to help you evaluate the given integral. $$ \int_{-\pi}^{\pi}(\sin x+\cos x) d x $$

Use the Interval Additive Property and linearity to evaluate \(\int_{0}^{4} f(x) d x .\) Begin by drawing a graph of \(f\). $$f(x)=|x-2|$$

use the Substitution Rule for Definite Integrals to evaluate each definite integral. $$ \int_{0}^{1}\left(x^{2}+1\right)^{10}(2 x) d x $$

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, evaluate the 10 -subinterval Riemann sums using left end point, right end point, and midpoint evaluations. $$ \int_{0}^{2}\left(x^{3}+1\right) d x $$

Use symmetry to help you evaluate the given integral. $$ \int_{-1}^{1} \frac{x^{3}}{\left(1+x^{2}\right)^{4}} d x $$

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

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