Chapter 5

Find the value of the indicated sum. $$ \sum_{k=1}^{6}(k-1) $$

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

In Problems 1-14, find the average value of the function on the given interval. 1\. \(f(x)=4 x^{3}, \quad[1,3]\)

In Problems 1-6, use the methods of (1) left Riemann sum, (2) right Riemann sum, (3) Trapezoidal Rule, (4) Parabolic Rule with \(n=8\) to approximate the definite integral. Then use the Second Fundamental Theorem of Calculus to find the exact value of each integral. $$ \int_{1}^{3} \frac{1}{x^{2}} d x $$

Find the value of the indicated sum. $$ \sum_{i=1}^{6} i^{-2} $$

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

In Problems 1-6, use the methods of (1) left Riemann sum, (2) right Riemann sum, (3) Trapezoidal Rule, (4) Parabolic Rule with \(n=8\) to approximate the definite integral. Then use the Second Fundamental Theorem of Calculus to find the exact value of each integral. $$ \int_{1}^{3} \frac{1}{x^{3}} d x $$

In Problems 3-6, calculate the Riemann sum \(\sum_{i=1}^{n} f\left(\bar{x}_{i}\right) \Delta x_{i}\) for the given data. \(f(x)=x-1 ; P: 3<3.75<4.25<5.5<6<7\); \(\bar{x}_{1}=3, \bar{x}_{2}=4, \bar{x}_{3}=4.75, \bar{x}_{4}=6, \bar{x}_{5}=6.5\)

Find the value of the indicated sum. $$ \sum_{k=1}^{7} \frac{1}{k+1} $$

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

\(f(x)=\frac{x}{\sqrt{x^{2}+16}} ;[0,3]\)

In Problems 1-6, use the methods of (1) left Riemann sum, (2) right Riemann sum, (3) Trapezoidal Rule, (4) Parabolic Rule with \(n=8\) to approximate the definite integral. Then use the Second Fundamental Theorem of Calculus to find the exact value of each integral. $$ \int_{0}^{2} \sqrt{x} d x $$

In Problems 3-6, calculate the Riemann sum \(\sum_{i=1}^{n} f\left(\bar{x}_{i}\right) \Delta x_{i}\) for the given data. \(f(x)=-x / 2+3 ; P:-3<-1.3<0<0.9<2\); \(\bar{x}_{1}=-2, \bar{x}_{2}=-0.5, \bar{x}_{3}=0, \bar{x}_{4}=2\)

Find the value of the indicated sum. $$ \sum_{l=3}^{8}(l+1)^{2} $$

In Problems 1-14, use the Second Fundamental Theorem of Calculus to evaluate each definite integral. \(\int_{1}^{2}\left(4 x^{3}+7\right) d x\)

\(f(x)=\frac{x^{2}}{\sqrt{x^{3}+16}}, \quad[0,2]\)

In Problems 1-6, use the methods of (1) left Riemann sum, (2) right Riemann sum, (3) Trapezoidal Rule, (4) Parabolic Rule with \(n=8\) to approximate the definite integral. Then use the Second Fundamental Theorem of Calculus to find the exact value of each integral. $$ \int_{1}^{3} x \sqrt{x^{2}+1} d x $$

Find the value of the indicated sum. $$ \sum_{m=1}^{8}(-1)^{m} 2^{m-2} $$

In Problems 1-14, use the Second Fundamental Theorem of Calculus to evaluate each definite integral. \(\int_{1}^{4} \frac{1}{w^{2}} d w\)

In Problems 1-6, use the methods of (1) left Riemann sum, (2) right Riemann sum, (3) Trapezoidal Rule, (4) Parabolic Rule with \(n=8\) to approximate the definite integral. Then use the Second Fundamental Theorem of Calculus to find the exact value of each integral. $$ \int_{0}^{1} x\left(x^{2}+1\right)^{5} d x $$

In Problems 3-6, calculate the Riemann sum \(\sum_{i=1}^{n} f\left(\bar{x}_{i}\right) \Delta x_{i}\) for the given data. \(f(x)=4 x^{3}+1 ;[0,3]\) is divided into six equal subintervals, \(\bar{x}_{i}\) is the right end point.

Find the value of the indicated sum. $$ \sum_{k=3}^{7} \frac{(-1)^{k} 2^{k}}{(k+1)} $$

In Problems 1-14, use the Second Fundamental Theorem of Calculus to evaluate each definite integral. \(\int_{1}^{3} \frac{2}{t^{3}} d t\)

In Problems 1-6, use the methods of (1) left Riemann sum, (2) right Riemann sum, (3) Trapezoidal Rule, (4) Parabolic Rule with \(n=8\) to approximate the definite integral. Then use the Second Fundamental Theorem of Calculus to find the exact value of each integral. $$ \int_{1}^{4}(x+1)^{3 / 2} d x $$

In Problems 7-10, use the given values of a and b and express the given limit as a definite integral. \(\lim _{|P| \rightarrow 0} \sum_{i=1}^{n}\left(\bar{x}_{i}\right)^{3} \Delta x_{i} ; a=1, b=3\)

Find the value of the indicated sum. $$ \sum_{n=1}^{6} n \cos (n \pi) $$

In Problems 1-14, use the Second Fundamental Theorem of Calculus to evaluate each definite integral. \(\int_{0}^{4} \sqrt{t} d t\)

In Problems 7-10, use the methods of (1) left Riemann sum, (2) right Riemann sum, (3) midpoint Riemann sum, (4) Trapezoidal Rule, (5) Parabolic Rule with \(n=4,8,16 .\) Present your approximations in a table like this: $$ \int_{1}^{3} \frac{1}{1+x^{2}} d x $$

In Problems 7-10, use the given values of a and b and express the given limit as a definite integral. \(\lim _{|P| \rightarrow 0} \sum_{i=1}^{n}\left(\bar{x}_{i}+1\right)^{3} \Delta x_{i} ; a=0, b=2\)

Find the value of the indicated sum. $$ \sum_{k=-1}^{6} k \sin (k \pi / 2) $$

In Problems 1-14, use the Second Fundamental Theorem of Calculus to evaluate each definite integral. \(\int_{1}^{8} \sqrt[3]{w} d w\)

In Problems 7-10, use the methods of (1) left Riemann sum, (2) right Riemann sum, (3) midpoint Riemann sum, (4) Trapezoidal Rule, (5) Parabolic Rule with \(n=4,8,16 .\) Present your approximations in a table like this: $$ \int_{1}^{3} \frac{1}{x} d x $$

In Problems 7-10, use the given values of a and b and express the given limit as a definite integral. \(\lim _{|P| \rightarrow 0} \sum_{i=1}^{n} \frac{\bar{x}_{i}^{2}}{1+\bar{x}_{i}} \Delta x_{i} ; a=-1, b=1\)

Write the indicated sum in sigma notation. $$ 1+2+3+\cdots+41 $$

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

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. $$ \int_{1}^{2} 2 f(x) d x $$

In Problems 7-10, use the given values of a and b and express the given limit as a definite integral. \(\lim _{\| P \mid \rightarrow 0} \sum_{i=1}^{n}\left(\sin \bar{x}_{i}\right)^{2} \Delta x_{i} ; a=0, b=\pi\)

Write the indicated sum in sigma notation. $$ 2+4+6+8+\cdots+50 $$

In Problems 1-14, use the Second Fundamental Theorem of Calculus to evaluate each definite integral. \(\int_{1}^{4} \frac{s^{4}-8}{s^{2}} d s\)

In Problems 7-10, use the methods of (1) left Riemann sum, (2) right Riemann sum, (3) midpoint Riemann sum, (4) Trapezoidal Rule, (5) Parabolic Rule with \(n=4,8,16 .\) Present your approximations in a table like this: $$ \int_{1}^{3} \ln \left(x^{2}+1\right) d x $$

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. $$ \int_{0}^{2} 2 f(x) d x $$

In Problems 11-16, evaluate the definite integrals using the definition, as in Examples 3 and \(4 .\) \(\int_{0}^{2}(x+1) d x\)

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

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

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

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

In Problems 11-16, evaluate the definite integrals using the definition, as in Examples 3 and \(4 .\) \(\int_{0}^{2}\left(x^{2}+1\right) d x\) Hint: Use \(\bar{x}_{i}=2 i / n\).

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

In Problems 1-14, use the Second Fundamental Theorem of Calculus to evaluate each definite integral. \(\int_{\pi / 6}^{\pi / 2} 2 \sin t d t\)

\(g(x)=\tan x \sec ^{2} x ; \quad[0, \pi / 4]\)