Chapter 5

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

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 \(n \rightarrow \infty .\) (See the example for \(y=x^{2}\) in the text. \()\) $$ y=x+2 ; a=0, b=1 $$

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 \(\bar{u} \leq \bar{v}\).

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

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 \(n \rightarrow \infty .\) (See the example for \(y=x^{2}\) in the text. \()\) $$ y=\frac{1}{2} x^{2}+1 ; a=0, b=1 $$

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 \(V_{\mathrm{rms}}\) is defined to be the square root of the average of \(V^{2} .\) Hence $$ V_{\mathrm{rms}}=\sqrt{\int_{\phi}^{1+\phi}(\hat{V} \sin (120 \pi t+\phi))^{2} d t} $$ A good measure of how much heat a given voltage can produce is given by \(V_{\mathrm{rms}}\) (a) Compute the average voltage over 1 second. (b) Compute the average voltage over \(1 / 60\) of a second. (c) Show that \(V_{\mathrm{rms}}=\frac{\hat{V} \sqrt{2}}{2}\) by computing the integral for \(V_{\mathrm{rms}}\) Hint: \(\int \sin ^{2} t d t=-\frac{1}{2} \cos t \sin t+\frac{1}{2} t+C .\) (d) If the \(V_{\mathrm{rms}}\) for household current is usually 120 volts, what is the value \(\hat{V}\) in this case?

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

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

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 \(n \rightarrow \infty .\) (See the example for \(y=x^{2}\) in the text. \()\) $$ y=2 x+2 ; a=-1, b=1 $$

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.

, use the Substitution Rule for Definite Integrals to evaluate each definite integral. $$ \int_{0}^{\pi} \sin x e^{\cos x} d x $$

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

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 \(\int_{0}^{2 \pi} \cos ^{2} x d x=\int_{0}^{2 \pi} \sin ^{2} x d x=\pi\).

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

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

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 \(n \rightarrow \infty .\) (See the example for \(y=x^{2}\) in the text. \()\) $$ y=x^{3} ; a=0, b=1 $$

Let \(f(x)=|\sin x| \sin (\cos x)\). (a) Is \(f\) even, odd, or neither? (b) Note that \(f\) is periodic. What is its period? (c) Evaluate the definite integral of \(f\) for each of the following intervals: \([0, \pi / 2],[-\pi / 2, \pi / 2],[0,3 \pi / 2],[-3 \pi / 2,3 \pi / 2]\) \([0,2 \pi],[\pi / 6,13 \pi / 6],[\pi / 6,4 \pi / 3],[13 \pi / 6,10 \pi / 3] .\)

, use the Substitution Rule for Definite Integrals to evaluate each definite integral.$$ \int_{0}^{1} x \cos ^{3}\left(x^{2}\right) \sin \left(x^{2}\right) d x $$

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

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 \(n \rightarrow \infty .\) (See the example for \(y=x^{2}\) in the text. \()\) $$ y=x^{3}+x ; a=0, b=1 $$

, use the Substitution Rule for Definite Integrals to evaluate each definite integral. $$ \int_{-\pi / 2}^{\pi / 2} x^{2} \sin ^{2}\left(x^{3}\right) \cos \left(x^{3}\right) d x $$

Decide whether the given statement is true or false. Then justify your answer. $$\text { If } \begin{aligned} \int_{a}^{b} f(x) d x &>\int_{a}^{b} g(x) d x, \text { then } \\ \int_{a}^{b}[f(x)-g(x)] d x>0 \end{aligned}$$

, use the Substitution Rule for Definite Integrals to evaluate each definite integral. $$ \int_{0}^{1} \frac{1}{1+x^{2}} d x $$

Decide whether the given statement is true or false. Then justify your answer. If \(f\) and \(g\) are continuous and \(f(x)>g(x)\) for all \(x\) in \([a, b]\), then \(\left|\int_{a}^{b} f(x) d x\right|>\left|\int_{a}^{b} g(x) d x\right| .\)

, use the Substitution Rule for Definite Integrals to evaluate each definite integral. $$ \int_{-1}^{1} x^{2} \cosh x^{3} d x $$

The velocity of an object is \(v(t)=2-|t-2|\). Assuming that the object is at the origin at time 0, find a formula for its position at time \(t\).

Let \(A_{a}^{b}\) denote the area under the curve \(y=x^{2}\) over the interval \([a, b]\). (a) Prove that \(A_{0}^{b}=b^{3} / 3 .\) Hint \(: \Delta x=b / n\), so \(x_{i}=i b / n ;\) use circumscribed polygons. (b) Show that \(A_{a}^{b}=b^{3} / 3-a^{3} / 3\). Assume that \(a \geq 0\).

The velocity of an object is $$v(t)=\left\\{\begin{array}{ll} 5 & \text { if } 0 \leq t \leq 100 \\ 6-t / 100 & \text { if } 100700 \end{array}\right.$$ (a) Assuming that the object is at the origin at time 0, find a formula for its position at time \(t(t \geq 0)\). (b) What is the farthest to the right of the origin that this object ever gets? (c) When, if ever, does the object return to the origin?

Suppose that an object, moving along the \(x\) -axis, has velocity \(v=t^{2}\) meters per second at time \(t\) seconds. How far did it travel between \(t=3\) and \(t=5\) ? See Problem 61 .

, use the Substitution Rule for Definite Integrals to evaluate each definite integral. $$ \int_{1}^{3} \frac{\ln x}{x} d x \text { Hint }: \text { Let } u=\ln x $$

Let \(f\) be continuous on \([a, b]\) and thus integrable there. Show that $$ \left|\int_{a}^{b} f(x) d x\right| \leq \int_{a}^{b}|f(x)| d x $$

Suppose that \(f^{\prime}\) is integrable and \(\left|f^{\prime}(x)\right| \leq M\) for all \(x\). Prove that \(|f(x)| \leq|f(a)|+M|x-a|\) for every \(a\).

. Water leaks out of a 200-gallon storage tank (initially full) at the rate \(V^{\prime}(t)=20-t\), where \(t\) is measured in hours and \(V\) in gallons. How much water leaked out between 10 and 20 hours? How long will it take the tank to drain completely?

Oil is leaking at the rate of \(V^{\prime}(t)=1-t / 110\) from a storage tank that is initially full of 55 gallons. How much leaks out during the first hour? During the tenth hour? How long until the entire tank is drained?

The mass, in kilograms, of a rod measured from the left endpoint to the point \(x\) meters away is \(m(x)=x+x^{2} / 8\). What is the density \(\delta(x)\) of the rod, measured in kilograms per meter? Assuming that the rod is 2 meters long, express the total mass of the rod in terms of its density.

first recognize the given limit as a definite integral and then evaluate that integral by the Second Fundamental Theorem of Calculus. $$ \lim _{n \rightarrow \infty} \sum_{i=1}^{n}\left(\frac{3 i}{n}\right)^{2} \frac{3}{n} $$

first recognize the given limit as a definite integral and then evaluate that integral by the Second Fundamental Theorem of Calculus. $$ \lim _{n \rightarrow \infty} \sum_{i=1}^{n}\left(\frac{2 i}{n}\right)^{3} \frac{2}{n} $$

first recognize the given limit as a definite integral and then evaluate that integral by the Second Fundamental Theorem of Calculus. $$ \lim _{n \rightarrow \infty} \sum_{i=1}^{n}\left[\sin \left(\frac{\pi i}{n}\right)\right] \frac{\pi}{n} $$

first recognize the given limit as a definite integral and then evaluate that integral by the Second Fundamental Theorem of Calculus. $$ \lim _{n \rightarrow \infty} \sum_{i=1}^{n}\left[1+\frac{2 i}{n}+\left(\frac{2 i}{n}\right)^{2}\right] \frac{2}{n} $$

Explain why \(\left(1 / n^{3}\right) \sum_{i=1}^{n} i^{2}\) should be a good approximation to \(\int_{0}^{1} x^{2} d x\) for large \(n .\) Now calculate the summation expression for \(n=10\), and evaluate the integral by the Second Fundamental Theorem of Calculus. Compare their values.

Show that \(\frac{1}{2} x|x|\) is an antiderivative of \(|x|\), and use this fact to get a simple formula for \(\int_{a}^{b}|x| d x\).

Give an example to show that the accumulation function \(G(x)=\int_{a}^{x} f(x) d x\) can be continuous even if \(f\) is not continuous.