Chapter 5

Suppose that \(\int_{0}^{1} f(x) d x=2, \int_{1}^{2} f(x) d x=3, \int_{0}^{1} g(x) d x=-1\), and \(\int_{0}^{2} g(x) d x=4\). Use properties of definite integrals (linearity, interval additivity, and so on) to calculate each of the integrals in Problems 9-16. $$ \int_{0}^{1}[2 f(s)+g(s)] d s $$

In Problems 11-16, evaluate the definite integrals using the definition, as in Examples 3 and \(4 .\) \(\int_{-2}^{1}(2 x+\pi) d x\)

Write the indicated sum in sigma notation. $$ a_{1}+a_{3}+a_{5}+a_{7}+\cdots+a_{99} $$

In Problems 1-14, use the Second Fundamental Theorem of Calculus to evaluate each definite integral. \(\int_{0}^{1}\left(2 x^{4}-3 x^{2}+5\right) d x\)

Suppose that \(\int_{0}^{1} f(x) d x=2, \int_{1}^{2} f(x) d x=3, \int_{0}^{1} g(x) d x=-1\), and \(\int_{0}^{2} g(x) d x=4\). Use properties of definite integrals (linearity, interval additivity, and so on) to calculate each of the integrals in Problems 9-16. $$ \int_{2}^{1}[2 f(s)+5 g(s)] d s $$

Write the indicated sum in sigma notation. $$ f\left(w_{1}\right) \Delta x+f\left(w_{2}\right) \Delta x+\cdots+f\left(w_{n}\right) \Delta x $$

In Problems 1-14, use the Second Fundamental Theorem of Calculus to evaluate each definite integral. \(\int_{0}^{1}\left(x^{4 / 3}-2 x^{1 / 3}\right) d x\)

\(G(v)=\frac{\sin v \cos v}{\sqrt{1+\cos ^{2} v}}, \quad[0, \pi / 2]\)

In Problems \(11-14\), determine an \(n\) so that the Trapezoidal Rule will approximate the integral with an error \(E_{n}\) satisfying \(\left|E_{n}\right| \leq 0.01\). Then, using that \(n\), approximate the integral. $$ \int_{1}^{3} e^{x} d x $$

Suppose that \(\int_{0}^{1} f(x) d x=2, \int_{1}^{2} f(x) d x=3, \int_{0}^{1} g(x) d x=-1\), and \(\int_{0}^{2} g(x) d x=4\). Use properties of definite integrals (linearity, interval additivity, and so on) to calculate each of the integrals in Problems 9-16. $$ \int_{1}^{1}[3 f(x)+2 g(x)] d x $$

Suppose that \(\sum_{i=1}^{10} a_{i}=40\) and \(\sum_{i=1}^{10} b_{i}=50\). Calculate each of the following. $$ \sum_{i=1}^{10}\left(a_{i}+b_{i}\right) $$

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

In Problems 15-28, find all values of \(c\) that satisfy the Mean Value Theorem for Integrals on the given interval. 15\. \(f(x)=\sqrt{x+1} ; \quad[0,3] \quad\)

Suppose that \(\int_{0}^{1} f(x) d x=2, \int_{1}^{2} f(x) d x=3, \int_{0}^{1} g(x) d x=-1\), and \(\int_{0}^{2} g(x) d x=4\). Use properties of definite integrals (linearity, interval additivity, and so on) to calculate each of the integrals in Problems 9-16. $$ \int_{0}^{2}[3 f(t)+2 g(t)] d t $$

Suppose that \(\sum_{i=1}^{10} a_{i}=40\) and \(\sum_{i=1}^{10} b_{i}=50\). Calculate each of the following. $$ \sum_{n=1}^{10}\left(3 a_{n}+2 b_{n}\right) $$

In Problems 15-34, use the method of substitution to find each of the following indefinite integrals. \(\int \sqrt[3]{2 x-4} d x\)

Suppose that \(\int_{0}^{1} f(x) d x=2, \int_{1}^{2} f(x) d x=3, \int_{0}^{1} g(x) d x=-1\), and \(\int_{0}^{2} g(x) d x=4\). Use properties of definite integrals (linearity, interval additivity, and so on) to calculate each of the integrals in Problems 9-16. $$ \int_{0}^{2}[\sqrt{3} f(t)+\sqrt{2} g(t)+\pi] d t $$

In Problems 17-22, 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)= \begin{cases}2 x & \text { if } 0 \leq x \leq 1 \\ 2 & \text { if } 1

Suppose that \(\sum_{i=1}^{10} a_{i}=40\) and \(\sum_{i=1}^{10} b_{i}=50\). Calculate each of the following. $$ \sum_{p=0}^{9}\left(a_{p+1}-b_{p+1}\right) $$

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

In Problems 17-26, find \(G^{\prime}(x)\). $$ G(x)=\int_{1}^{x} 2 t d t $$

In Problems 17-22, 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)= \begin{cases}3 x & \text { if } 0 \leq x \leq 1 \\ 2(x-1)+2 & \text { if } 1

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

In Problems 17-26, find \(G^{\prime}(x)\). $$ G(x)=\int_{x}^{1} 2 t d t $$

In Problems 17-22, 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)= \begin{cases}\sqrt{1-x^{2}} & \text { if } 0 \leq x \leq 1 \\ x-1 & \text { if } 1

In Problems 15-34, use the method of substitution to find each of the following indefinite integrals. \(\int \sin (6 x-7) d x\)

If a function \(f\) is increasing on \([a, b]\), will the left Riemann sum be larger or smaller than \(\int_{a}^{b} f(x) d x\) ?

In Problems 17-26, find \(G^{\prime}(x)\). $$ G(x)=\int_{0}^{x}\left(2 t^{2}+\sqrt{t}\right) d t $$

In Problems 17-22, 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)= \begin{cases}-\sqrt{4-x^{2}} & \text { if }-2 \leq x \leq 0 \\ -2 x-2 & \text { if } 0

In Problems 15-34, use the method of substitution to find each of the following indefinite integrals. \(\int \cos (\pi v-\sqrt{7}) d v\)

If a function \(f\) is increasing on \([a, b]\), will the right Riemann sum be larger or smaller than \(\int_{a}^{b} f(x) d x\) ?

In Problems 17-26, find \(G^{\prime}(x)\). $$ G(x)=\int_{1}^{x} \cos ^{3} 2 t \tan t d t ;-\pi / 2

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

If a function \(f\) is concave down on \([a, b]\), will the midpoint Riemann sum be larger or smaller than \(\int_{a}^{b} f(x) d x\) ?

In Problems 17-26, find \(G^{\prime}(x)\). $$ G(x)=\int_{x}^{\pi / 4}(s-2) \cot 2 s d s ; 0

In Problems 17-22, 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. \(f(x)=4-|x|,-4 \leq x \leq 4\)

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

\(\quad g(y)=\cos 2 y ; \quad[0, \pi]\)

If a function \(f\) is concave down on \([a, b]\), will the Trapezoidal Rule approximation be larger or smaller than \(\int_{a}^{b} f(x) d x\) ?

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

In Problems 15-34, use the method of substitution to find each of the following indefinite integrals. \(\int x\left(x^{2}+3\right)^{-12 / 7} d x\)

In Problems 17-26, find \(G^{\prime}(x)\). $$ G(x)=\int^{x^{2}} e^{-t^{2}} d t $$

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

In Problems 15-34, use the method of substitution to find each of the following indefinite integrals. \(\int v\left(\sqrt{3} v^{2}+\pi\right)^{7 / 8} d v\)

In Problems 17-26, find \(G^{\prime}(x)\). $$ G(x)=\int_{1}^{x^{2}+x} \sqrt{2 z+\sin z} d z $$

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}t / 2 & \text { if } 0 \leq t \leq 2 \\ 1 & \text { if } 2

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

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

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.

In Problems 17-26, find \(G^{\prime}(x)\). $$ G(x)=\int_{-x^{2}}^{x} \frac{t^{2}}{1+t^{2}} d t \text { Hint: } \int_{-x^{2}}^{x}=\int_{-x^{2}}^{0}+\int_{0}^{x} $$