Chapter 5

A function \(f,\) an interval \(I,\) and an even integer \(N\) are given. Approximate the integral of \(f\) over \(I\) by partitioning \(I\) into \(N\) equal length subintervals and using the Midpoint Rule, the Trapezoidal Rule, and then Simpson's Rule. $$ f(x)=x^{2} \quad I=[1,3], N=2 $$

A function \(f\) is defined on a specified interval \(I=[a, b] .\) Calculate the area of the region that lies between the vertical lines \(x=a\) and \(x=b\) and between the graph of \(f\) and the \(x\) -axis. \(f(x)=2 \cos (x) \quad I=[\pi / 4,2 \pi / 3]\)

Determine a substitution that will simplify the integral. In each problem, record your choice of \(u\) and the resulting expression for \(d u .\) Then evaluate the integral. $$ \int \sin (3 x) d x $$

Suppose that \(\int_{1}^{3} f(x) d x=-8\) and \(\int_{3}^{7} f(x) d x=12\). Evaluate \(\int_{1}^{7} f(x) d x\)

P In Exercises \(1-10\), write out the sum, and perform the addition. $$ \sum_{j=1}^{4} 3 j $$

For the given function \(f,\) interval \(I,\) and uniform partition of order \(N\) : a. Evaluate the Riemann sum \(\mathcal{R}(f, \mathcal{S})\) using the choice of points \(\mathcal{S}\) that consists of the midpoints of the subintervals of \(I\). b. Evaluate the definite integral that \(\mathcal{R}(f, \mathcal{S})\) approximates. \(f(x)=1 / x \quad \mathcal{S}=\\{4,8,12\\}, I=[2,14], N=3\)

A function \(f,\) an interval \(I,\) and an even integer \(N\) are given. Approximate the integral of \(f\) over \(I\) by partitioning \(I\) into \(N\) equal length subintervals and using the Midpoint Rule, the Trapezoidal Rule, and then Simpson's Rule. $$ f(x)=x^{1 / 2} \quad I=[0,2], N=2 $$

A function \(f\) is defined on a specified interval \(I=[a, b] .\) Calculate the area of the region that lies between the vertical lines \(x=a\) and \(x=b\) and between the graph of \(f\) and the \(x\) -axis. \(f(x)=3 x^{2}-3 x-6 \quad I=[-2,4]\)

Determine a substitution that will simplify the integral. In each problem, record your choice of \(u\) and the resulting expression for \(d u .\) Then evaluate the integral. $$ \int 24 \sec ^{2}(4 t) d t $$

Use the first part of the Fundamental Theorem of Calculus together with the ideas of Examples 1 and 2 to evaluate the definite integrals in Exercises \(1-10\) $$ \int_{-1}^{0} e^{-x} d x $$

Suppose that \(\int_{2}^{12} g(x) d x=-6\) and \(\int_{2}^{6} g(x) d x=-12\). Evaluate \(\int_{6}^{12} g(x) d x\).

P In Exercises \(1-10\), write out the sum, and perform the addition. $$ \sum_{j=0}^{6}(2 j-1) $$

A function \(f,\) an interval \(I,\) and an even integer \(N\) are given. Approximate the integral of \(f\) over \(I\) by partitioning \(I\) into \(N\) equal length subintervals and using the Midpoint Rule, the Trapezoidal Rule, and then Simpson's Rule. $$ f(x)=\sqrt{2+x+x^{2}} \quad I=[-1,1], N=2 $$

A function \(f\) is defined on a specified interval \(I=[a, b] .\) Calculate the area of the region that lies between the vertical lines \(x=a\) and \(x=b\) and between the graph of \(f\) and the \(x\) -axis. \(f(x)=2 x^{2}-8 \quad I=[-3,5]\)

Determine a substitution that will simplify the integral. In each problem, record your choice of \(u\) and the resulting expression for \(d u .\) Then evaluate the integral. $$ \int 64\left(x^{8}+1\right)^{-5} x^{7} d x $$

Suppose that \(\int_{-7}^{3} f(x) d x=-7\) and \(\int_{-7}^{3} g(x) d x=-4\). Evaluate \(\int_{-7}^{3}(4 f(x)-9 g(x)) d x\).

P In Exercises \(1-10\), write out the sum, and perform the addition. $$ \sum_{j=2}^{5}\left(-2 j^{2}\right) $$

A function \(f,\) an interval \(I,\) and an even integer \(N\) are given. Approximate the integral of \(f\) over \(I\) by partitioning \(I\) into \(N\) equal length subintervals and using the Midpoint Rule, the Trapezoidal Rule, and then Simpson's Rule. $$ f(x)=\left(1+x^{3}\right)^{(1 / 3)} \quad I=[-1,1], N=2 $$

A function \(f\) is defined on a specified interval \(I=[a, b] .\) Calculate the area of the region that lies between the vertical lines \(x=a\) and \(x=b\) and between the graph of \(f\) and the \(x\) -axis. \(f(x)=4 / x-x \quad \quad I=[1,3]\)

Determine a substitution that will simplify the integral. In each problem, record your choice of \(u\) and the resulting expression for \(d u .\) Then evaluate the integral. $$ \int 24 t \sqrt{t^{2}+4} d t $$

Suppose that \(\int_{4}^{8} f(x) d x=6 .\) Evaluate \(\int_{8}^{4} f(x) d x\).

P In Exercises \(1-10\), write out the sum, and perform the addition. $$ \sum_{\ell=4}^{6} \ell /(\ell-3) $$

A function \(f,\) an interval \(I,\) and an even integer \(N\) are given. Approximate the integral of \(f\) over \(I\) by partitioning \(I\) into \(N\) equal length subintervals and using the Midpoint Rule, the Trapezoidal Rule, and then Simpson's Rule. $$ f(x)=\ln (x) \quad I=[1,3], N=2 $$

A function \(f\) is defined on a specified interval \(I=[a, b] .\) Calculate the area of the region that lies between the vertical lines \(x=a\) and \(x=b\) and between the graph of \(f\) and the \(x\) -axis. \(f(x)=x^{3}+x+2 \quad I=[-2,1]\)

Determine a substitution that will simplify the integral. In each problem, record your choice of \(u\) and the resulting expression for \(d u .\) Then evaluate the integral. $$ \int 30 x^{2}\left(x^{3}-5\right)^{3 / 2} d x $$

Suppose that \(\int_{7}^{-2} f(x) d x=6\) and \(\int_{7}^{9} f(x) d x=-4 .\) Evaluate \(\int_{9}^{-2} f(x) d x\)

P In Exercises \(1-10\), write out the sum, and perform the addition. $$ \sum_{n=2}^{5} 2 n /(n-1) $$

A function \(f,\) an interval \(I,\) and an even integer \(N\) are given. Approximate the integral of \(f\) over \(I\) by partitioning \(I\) into \(N\) equal length subintervals and using the Midpoint Rule, the Trapezoidal Rule, and then Simpson's Rule. $$ f(x)=15 x /(1+x) \quad I=[0,4], N=4 $$

A function \(f\) is defined on a specified interval \(I=[a, b] .\) Calculate the area of the region that lies between the vertical lines \(x=a\) and \(x=b\) and between the graph of \(f\) and the \(x\) -axis. \(f(x)=x /\left(x^{2}+1\right)^{2} \quad I=[-1,3]\)

Determine a substitution that will simplify the integral. In each problem, record your choice of \(u\) and the resulting expression for \(d u .\) Then evaluate the integral. $$ \int(\sqrt{t}+4)^{6} t^{-1 / 2} d t $$

Suppose that \(\int_{-3}^{-7} g(x) d x=5\) and \(\int_{-3}^{-5} g(x) d x=12 .\) Evaluate \(\int_{-7}^{-5} g(x) d x\).

P In Exercises \(1-10\), write out the sum, and perform the addition. $$ \sum_{k=2}^{4}\left(k^{3}-6 k\right) $$

A function \(f,\) an interval \(I,\) and an even integer \(N\) are given. Approximate the integral of \(f\) over \(I\) by partitioning \(I\) into \(N\) equal length subintervals and using the Midpoint Rule, the Trapezoidal Rule, and then Simpson's Rule. $$ f(x)=\cos (x) \quad I=[\pi / 4,5 \pi / 4], N=4 $$

A function \(f\) is defined on a specified interval \(I=[a, b] .\) Calculate the area of the region that lies between the vertical lines \(x=a\) and \(x=b\) and between the graph of \(f\) and the \(x\) -axis. \(f(x)=8 x \cdot\left(1-x^{2}\right)^{2} \quad I=[-1 / 2,1]\)

Determine a substitution that will simplify the integral. In each problem, record your choice of \(u\) and the resulting expression for \(d u .\) Then evaluate the integral. $$ \int \sin (s) \cos ^{4}(s) d s $$

For the given function \(f,\) interval \(I,\) and uniform partition of order \(N\) : a. Evaluate the Riemann sum \(\mathcal{R}(f, \mathcal{S})\) using the choice of points \(\mathcal{S}\) that consists of the left endpoints of the subintervals of \(I\). b. Evaluate the definite integral that \(\mathcal{R}(f, \mathcal{S})\) approximates. $$ f(x)=|x| \quad \mathcal{S}=\\{-7,-4,-1,2\\}, I=[-7,5], N=4 $$

Use the first part of the Fundamental Theorem of Calculus together with the ideas of Examples 1 and 2 to evaluate the definite integrals in Exercises \(1-10\) $$ \int_{\pi / 2}^{\pi} \sec (x / 3) \tan (x / 3) d x $$

Suppose that \(\int_{9}^{4} f(x) d x=5\) and \(\int_{9}^{4} g(x) d x=15 .\) Evaluate \(\int_{4}^{9}(6 f(x)-7 g(x)) d x\).

P In Exercises \(1-10\), write out the sum, and perform the addition. $$ \sum_{m=3}^{6}\left(2 m^{2}-3 m\right) $$

A function \(f,\) an interval \(I,\) and an even integer \(N\) are given. Approximate the integral of \(f\) over \(I\) by partitioning \(I\) into \(N\) equal length subintervals and using the Midpoint Rule, the Trapezoidal Rule, and then Simpson's Rule. $$ f(x)=\sqrt{1+x} \quad I=[0,8], N=4 $$

A function \(f\) is defined on a specified interval \(I=[a, b] .\) Calculate the area of the region that lies between the vertical lines \(x=a\) and \(x=b\) and between the graph of \(f\) and the \(x\) -axis. $$ f(x)=\exp (x)-e \quad I=[-1,2] $$

Determine a substitution that will simplify the integral. In each problem, record your choice of \(u\) and the resulting expression for \(d u .\) Then evaluate the integral. $$ \int 24 \frac{\sqrt{t}+1}{\sqrt{t}} d t $$

For the given function \(f,\) interval \(I,\) and uniform partition of order \(N\) : a. Evaluate the Riemann sum \(\mathcal{R}(f, \mathcal{S})\) using the choice of points \(\mathcal{S}\) that consists of the left endpoints of the subintervals of \(I\). b. Evaluate the definite integral that \(\mathcal{R}(f, \mathcal{S})\) approximates. $$ f(x)=1 / x \quad \mathcal{S}=\\{1,5 / 4.3 / 2,7 / 4,2\\}, I=[1,9 / 4], N=5 $$

Suppose that \(\int_{3}^{5} f(x) d x=2\). Evaluate \(\int_{5}^{3}-4 f(x) d x\).

P In Exercises \(1-10\), write out the sum, and perform the addition. $$ \sum_{j=1}^{5} j \cdot \sin (j \pi / 2) $$

A function \(f,\) an interval \(I,\) and an even integer \(N\) are given. Approximate the integral of \(f\) over \(I\) by partitioning \(I\) into \(N\) equal length subintervals and using the Midpoint Rule, the Trapezoidal Rule, and then Simpson's Rule. $$ f(x)=x^{3} \quad I=[-1 / 2,7 / 2], N=4 $$

A function \(f\) is defined on a specified interval \(I=[a, b] .\) Calculate the area of the region that lies between the vertical lines \(x=a\) and \(x=b\) and between the graph of \(f\) and the \(x\) -axis. $$ f(x)=x\left(4-x^{2}\right)^{3} \quad I=[-2,3] $$

Determine a substitution that will simplify the integral. In each problem, record your choice of \(u\) and the resulting expression for \(d u .\) Then evaluate the integral. $$ \int \frac{\sin (\pi / t)}{t^{2}} d t $$

For the given function \(f\), interval \(I\), and uniform partition of order \(N\) : a. Evaluate the Riemann sum \(\mathcal{R}(f, \mathcal{S})\) using the choice of points \(\mathcal{S}\) that consists of the right endpoints of the subintervals of \(I\). b. Evaluate the definite integral that \(\mathcal{R}(f, \mathcal{S})\) approximates. $$ f(x)=e^{x} \quad \mathcal{S}=\\{-2,0,2\\}, I=[-4,2], N=3 $$

Suppose that \(\int_{5}^{-3}(-3 f(x) / 4) d x=7\). Evaluate \(\int_{5}^{-3}(6 f(x)+1) d x\).