Chapter 5

\(\int_{-1}^{1}\left(1+x+x^{2}+x^{3}\right) d x\)

Show that \(1 \leq \int_{0}^{1} \sqrt{1+x^{4}} d x \leq \frac{6}{5}\). Hint: Explain why \(1 \leq \sqrt{1+x^{4}} \leq 1+x^{4}\) for \(x\) in the closed interval \([0,1]\); then use the Comparison Property (Theorem B) and the result of Problem 39d.

Find a nice formula for the sum $$ \frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+\frac{1}{3 \cdot 4}+\cdots+\frac{1}{n(n+1)} $$

In Problems 35-62, use the Substitution Rule for Definite Integrals to evaluate each definite integral. \(\int_{1}^{3} \frac{x^{2}+1}{\sqrt{x^{3}+3 x}} d x\)

\(\int_{-100}^{100}\left(v+\sin v+v \cos v+\sin ^{3} v\right)^{5} d v\)

In Problems 35-62, use the Substitution Rule for Definite Integrals to evaluate each definite integral. \(\int_{0}^{\pi / 2} \cos ^{2} x \sin x d x\)

\(\int_{-1}^{1} x e^{-4 x^{2}} d x\)

In Problems 43-48, use a graphing calculator to graph each integrand. Then use the Boundedness Property (Theorem \(C\) ) to find a lower bound and an upper bound for each definite integral. $$ \int_{0}^{4}\left(5+x^{3}\right) d x $$

In Problems 35-62, use the Substitution Rule for Definite Integrals to evaluate each definite integral. \(\int_{0}^{\pi / 2} \sin ^{2} 3 x \cos 3 x d x\)

\(\int_{-\pi / 4}^{\pi / 4}\left(|x| \sin ^{5} x+|x|^{2} \tan x\right) d x\)

In Problems 35-62, use the Substitution Rule for Definite Integrals to evaluate each definite integral. \(\int_{0}^{1} x e^{x^{2}} d x\)

How does \(\int_{-b}^{-a} f(x) d x\) compare with \(\int_{a}^{b} f(x) d x\) when \(f\) is an even function? An odd function?

In Problems 43-48, use a graphing calculator to graph each integrand. Then use the Boundedness Property (Theorem \(C\) ) to find a lower bound and an upper bound for each definite integral. $$ \int_{1}^{5}\left(3+\frac{2}{x}\right) d x $$

In Problems 35-62, use the Substitution Rule for Definite Integrals to evaluate each definite integral. \(\int_{1}^{4} \frac{(\sqrt{x}-1)^{3}}{\sqrt{x}} d x\)

Prove (by a substitution) that $$ \int_{a}^{b} f(-x) d x=\int_{-b}^{-a} f(x) d x $$

In Problems 43-48, use a graphing calculator to graph each integrand. Then use the Boundedness Property (Theorem \(C\) ) to find a lower bound and an upper bound for each definite integral. $$ \int_{10}^{20}\left(1+\frac{1}{x}\right)^{5} d x $$

In Problems 35-62, use the Substitution Rule for Definite Integrals to evaluate each definite integral. \(\int_{0}^{\pi / 6} \sin ^{3} \theta \cos \theta d \theta\)

Use periodicity to calculate \(\int_{0}^{4 \pi}|\cos x| d x\).

In Problems 43-48, use a graphing calculator to graph each integrand. Then use the Boundedness Property (Theorem \(C\) ) to find a lower bound and an upper bound for each definite integral. $$ \int_{4 \pi}^{8 \pi}\left(5+\frac{1}{20} \sin ^{2} x\right) d x $$

In Problems 35-62, use the Substitution Rule for Definite Integrals to evaluate each definite integral. \(\int_{0}^{\pi / 6} \frac{\sin \theta}{\cos ^{3} \theta} d \theta\)

Calculate \(\int_{0}^{4 \pi}|\sin 2 x| d x\).

In Problems 43-48, use a graphing calculator to graph each integrand. Then use the Boundedness Property (Theorem \(C\) ) to find a lower bound and an upper bound for each definite integral. $$ \int_{0.2}^{0.4}\left(0.002+0.0001 \cos ^{2} x\right) d x $$

Sketch the graph of the given function over the interval \([a, b]\); then divide \([a, b]\) into \(n\) equal subintervals. Finally, calculate the area of the corresponding circumscribed polygon. $$ f(x)=x+1 ; a=-1, b=2, n=3 $$

In Problems 35-62, use the Substitution Rule for Definite Integrals to evaluate each definite integral. \(\int_{0}^{1} \cos (3 x-3) d x\)

If \(f\) is periodic with period \(p\), then $$ \int_{a}^{a+p} f(x) d x=\int_{0}^{p} f(x) d x $$ Convince yourself that this is true by drawing a picture and then use the result to calculate \(\int_{1}^{1+\pi}|\sin x| d x\).

In Problems 43-48, use a graphing calculator to graph each integrand. Then use the Boundedness Property (Theorem \(C\) ) to find a lower bound and an upper bound for each definite integral. $$ \text { Find } \lim _{x \rightarrow 0} \frac{1}{x} \int_{0}^{x} \frac{1+t}{2+t} d t $$

Sketch the graph of the given function over the interval \([a, b]\); then divide \([a, b]\) into \(n\) equal subintervals. Finally, calculate the area of the corresponding circumscribed polygon. $$ f(x)=3 x-1 ; a=1, b=3, n=4 $$

In Problems 35-62, use the Substitution Rule for Definite Integrals to evaluate each definite integral. \(\int_{0}^{1 / 2} \sin (2 \pi x) d x\)

Sketch the graph of the given function over the interval \([a, b]\); then divide \([a, b]\) into \(n\) equal subintervals. Finally, calculate the area of the corresponding circumscribed polygon. $$ f(x)=x^{2}-1 ; a=2, b=3, n=6 $$

In Problems 35-62, use the Substitution Rule for Definite Integrals to evaluate each definite integral. \(\int_{0}^{1} x \sin \left(\pi x^{2}\right) d x\)

Calculate \(\int_{1}^{1+\pi}|\cos x| d x\).

Sketch the graph of the given function over the interval \([a, b]\); then divide \([a, b]\) into \(n\) equal subintervals. Finally, calculate the area of the corresponding circumscribed polygon. $$ f(x)=3 x^{2}+x+1 ; a=-1, b=1, n=10 $$

In Problems 35-62, use the Substitution Rule for Definite Integrals to evaluate each definite integral. \(\int_{0}^{\pi} x^{4} \cos \left(2 x^{5}\right) d x\)

Prove or disprove that the integral of the average value equals the integral of the function on the interval: \(\int_{a}^{b} \bar{f} d x=\) \(\int_{a}^{b} f(x) d x\), where \(\bar{f}\) is the average value of the function \(f\) over the interval \([a, b]\).

In Problems 43-48, use a graphing calculator to graph each integrand. Then use the Boundedness Property (Theorem \(C\) ) to find a lower bound and an upper bound for each definite integral. $$ \text { Find } f(x) \text { if } \int_{0}^{x} f(t) d t=x^{2} $$

Find the area of the region under the curve \(y=f(x)\) over the interval \([a, b]\). To do this, divide the interval \([a, b]\) into n equal subintervals, calculate the area of the corresponding circumscribed polygon, and then let \(n \rightarrow \infty\). $$ y=x+2 ; a=0, b=1 $$

In Problems 35-62, use the Substitution Rule for Definite Integrals to evaluate each definite integral. \(\int_{0}^{\pi / 4}(\cos 2 x+\sin 2 x) d x\)

Assuming that \(u\) and \(v\) can be integrated over the interval \([a, b]\) and that the average values over the interval are denoted by \(\bar{u}\) and \(\bar{v}\), prove or disprove that (a) \(\bar{u}+\bar{v}=\overline{u+v}\), (b) \(k \bar{u}=\overline{k u}\), where \(k\) is any constant; (c) if \(u \leq v\) then \(\bar{u} \leq \bar{v}\).

In Problems 43-48, use a graphing calculator to graph each integrand. Then use the Boundedness Property (Theorem \(C\) ) to find a lower bound and an upper bound for each definite integral. $$ \text { Find } f(x) \text { if } \int_{0}^{x^{2}} f(t) d t=\frac{1}{3} x^{3} $$

Find the area of the region under the curve \(y=f(x)\) over the interval \([a, b]\). To do this, divide the interval \([a, b]\) into n equal subintervals, calculate the area of the corresponding circumscribed polygon, and then let \(n \rightarrow \infty\). $$ y=\frac{1}{2} x^{2}+1 ; a=0, b=1 $$

In Problems 35-62, use the Substitution Rule for Definite Integrals to evaluate each definite integral. \(\int_{-\pi / 2}^{\pi / 2}(\cos 3 x+\sin 5 x) d x\)

Household electric current can be modeled by the voltage \(V=\hat{V} \sin (120 \pi t+\phi)\), where \(t\) is measured in seconds, \(\hat{V}\) is the maximum value that \(V\) can attain, and \(\phi\) is the phase angle. Such a voltage is usually said to be 60 -cycle, since in 1 second the voltage goes through 60 oscillations. The root-mean-square voltage, usually denoted by \(V_{\text {rms }}\) is defined to be the square root of the average of \(V^{2}\). Hence $$ V_{\mathrm{rmss}}=\sqrt{\int_{\phi}^{1+\phi}(\hat{V} \sin (120 \pi t+\phi))^{2} d t} $$ A good measure of how much heat a given voltage can produce is given by \(V_{\text {rms }}\). (a) Compute the average voltage over 1 second. (b) Compute the average voltage over \(1 / 60\) of a second. (c) Show that \(V_{r m s}=\frac{\hat{V} \sqrt{2}}{2}\) by computing the integral for \(V_{\text {rms. }}\). Hint: \(\int \sin ^{2} t d t=-\frac{1}{2} \cos t \sin t+\frac{1}{2} t+C\). (d) If the \(V_{\text {rms }}\) for household current is usually 120 volts, what is the value \(\hat{V}\) in this case?

Does there exist a function \(f\) such that \(\int_{0}^{x} f(t) d t=\) \(x+1\) ? Explain.

Find the area of the region under the curve \(y=f(x)\) over the interval \([a, b]\). To do this, divide the interval \([a, b]\) into n equal subintervals, calculate the area of the corresponding circumscribed polygon, and then let \(n \rightarrow \infty\). $$ y=2 x+2 ; a=-1, b=1 . \text { Hint: } x_{i}=-1+\frac{2 u}{n} $$

In Problems 35-62, use the Substitution Rule for Definite Integrals to evaluate each definite integral. \(\int_{0}^{\pi} \sin x e^{\cos x} d x\)

Give a proof of the Mean Value Theorem for Integrals (Theorem A) that does not use the First Fundamental Theorem of Calculus. Hint: Apply the Max-Min Existence Theorem and the Intermediate Value Theorem. 56\. Integrals that occur frequently in applications are \(\int_{0}^{2 \pi} \cos ^{2} x d x\) and \(\int_{0}^{2 \pi} \sin ^{2} x d x\) (a) Using a trigonometric identity, show that $$ \int_{0}^{2 \pi}\left(\sin ^{2} x+\cos ^{2} x\right) d x=2 \pi $$ (b) Show from graphical considerations that $$ \int_{0}^{2 \pi} \cos ^{2} x d x=\int_{0}^{2 \pi} \sin ^{2} x d x $$ (c) Conclude that \(\int_{0}^{2 \pi} \cos ^{2} x d x=\int_{0}^{2 \pi} \sin ^{2} x d x=\pi\).

Decide whether the given statement is true or false. Then justify your answer. If \(f\) is continuous and \(f(x) \geq 0\) for all \(x\) in \([a, b]\), then \(\int^{b} f(x) d x \geq 0\)

Find the area of the region under the curve \(y=f(x)\) over the interval \([a, b]\). To do this, divide the interval \([a, b]\) into n equal subintervals, calculate the area of the corresponding circumscribed polygon, and then let \(n \rightarrow \infty\). $$ y=x^{2} ; a=-2, b=2 $$

In Problems 35-62, use the Substitution Rule for Definite Integrals to evaluate each definite integral. \(\int_{-\pi / 2}^{\pi / 2} \cos \theta \cos (\pi \sin \theta) d \theta\)

Integrals that occur frequently in applications are \(\int_{0}^{2 \pi} \cos ^{2} x d x\) and \(\int_{0}^{2 \pi} \sin ^{2} x d x\) (a) Using a trigonometric identity, show that $$ \int_{0}^{2 \pi}\left(\sin ^{2} x+\cos ^{2} x\right) d x=2 \pi $$ (b) Show from graphical considerations that $$ \int_{0}^{2 \pi} \cos ^{2} x d x=\int_{0}^{2 \pi} \sin ^{2} x d x $$ (c) Conclude that \(\int_{0}^{2 \pi} \cos ^{2} x d x=\int_{0}^{2 \pi} \sin ^{2} x d x=\pi\).