Chapter 5

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. $$ \int_{1}^{3} \frac{1}{x} d x $$

Find the average value of the function on the given interval. $$ F(y)=y\left(1+y^{2}\right)^{3} ; \quad[1,2] $$

use the Second Fundamental Theorem of Calculus to evaluate each definite integral. $$ \int_{0}^{\pi / 2} \cos x d x $$

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. $$ \int_{0}^{2}[2 f(x)+g(x)] d x $$

Write the indicated sum in sigma notation. $$ 1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots-\frac{1}{100} $$

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. $$ \int_{1}^{3} \frac{1}{1+x} d x $$

Find the average value of the function on the given interval. $$ g(x)=\tan x \sec ^{2} x ; \quad[0, \pi / 4] $$

use the Second Fundamental Theorem of Calculus to evaluate each definite integral. $$ \int_{\pi / 6}^{\pi / 2} 2 \sin t d t $$

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. $$ \int_{0}^{1}[2 f(s)+g(s)] d s $$

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

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. $$ \int_{1}^{4} \sqrt{x} d x $$

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. $$ \int_{2}^{1}[2 f(s)+5 g(s)] d s $$

Find the average value of the function on the given interval. $$ h(z)=\frac{\sin \sqrt{z}}{\sqrt{z}} ; \quad[\pi / 4, \pi / 2] $$

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

Evaluate the definite integrals using the definition, as in Examples 3 and \(4 .\) $$ \int_{-2}^{1}(2 x+\pi) d x $$

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. $$ \int_{1}^{1}[3 f(x)+2 g(x)] d x $$

Find the average value of the function on the given interval. $$ G(v)=\frac{\sin v \cos v}{\sqrt{1+\cos ^{2} v}} ; \quad[0, \pi / 2] $$

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

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

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. $$ \int_{0}^{2}[3 f(t)+2 g(t)] d t $$

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

use the method of substitution to find each of the following indefinite integrals. $$ \int \sqrt{3 x+2} d x $$

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

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. $$ \int_{0}^{2}[\sqrt{3} f(t)+\sqrt{2} g(t)+\pi] d t $$

Find all values of \(c\) that satisfy the Mean Value Theorem for Integrals on the given interval. $$ f(x)=x^{2} ; \quad[-1,1] $$

use the method of substitution to find each of the following indefinite integrals. $$ \int \sqrt[3]{2 x-4} d x $$

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

$$ \begin{array}{l} \text { Let } f(x)=a x^{2}+b x+c \text { . Show that }\\\ \int_{m-h}^{m+h} f(x) d x \text { and } \frac{h}{3}[f(m-h)+4 f(m)+f(m+h)] \end{array} $$

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

Find all values of \(c\) that satisfy the Mean Value Theorem for Integrals on the given interval. $$ f(x)=1-x^{2} ; \quad[-4,3] $$

use the method of substitution to find each of the following indefinite integrals. $$ \int \cos (3 x+2) d x $$

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)=\left\\{\begin{array}{ll} 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. $$ \sum_{q=1}^{10}\left(a_{q}-b_{q}-q\right) $$

Find all values of \(c\) that satisfy the Mean Value Theorem for Integrals on the given interval. $$ f(x)=x(1-x) ; \quad[0,1] $$

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

use the method of substitution to find each of the following indefinite integrals. $$ \int \sin (2 x-4) d x $$

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)=\left\\{\begin{array}{ll} 3 x & \text { if } 0 \leq x \leq 1 \\ 2(x-1)+2 & \text { if } 1

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

Find \(G^{\prime}(x)\). $$ G(x)=\int_{0}^{x}\left(2 t^{2}+\sqrt{t}\right) d t $$

use the method of substitution to find each of the following indefinite integrals. $$ \int \sin (6 x-7) d x $$

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)=\left\\{\begin{array}{ll} \sqrt{1-x^{2}} & \text { if } 0 \leq x \leq 1 \\ x-1 & \text { if } 1

Use Special Sum Formulas 1-4 to find each sum. $$ \sum_{i=1}^{10}[(i-1)(4 i+3)] $$

Find all values of \(c\) that satisfy the Mean Value Theorem for Integrals on the given interval. $$ f(x)=|x| ; \quad[-2,2] $$

Find \(G^{\prime}(x)\). $$ G(x)=\int_{1}^{x} \cos ^{3} 2 t \tan t d t ;-\pi / 2

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

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

Find all values of \(c\) that satisfy the Mean Value Theorem for Integrals on the given interval. $$ H(z)=\sin z ; \quad[-\pi, \pi] $$

Find \(G^{\prime}(x)\). $$ G(x)=\int_{x}^{\pi / 4}(s-2) \cot 2 s d s ; 0

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