Chapter 5

In the following exercises, use a calculator to estimate the area under the curve by computing \(T_{10}\) , the average of the left- and right-endpoint Riemann sums using \(N=10\) rectangles. Then, using the Fundamental Theorem of Calculus, Part 2 , determine the exact area. $$ y=\sqrt{x^{3}} \text { over }[0,6] $$

In the following exercises, use a calculator to estimate the area under the curve by computing \(T_{10}\) , the average of the left- and right-endpoint Riemann sums using \(N=10\) rectangles. Then, using the Fundamental Theorem of Calculus, Part 2 , determine the exact area. $$ y=\sqrt{x}+x^{2} \text { over }[1,9] $$

In the following exercises, use a calculator to estimate the area under the curve by computing \(T_{10}\) , the average of the left- and right-endpoint Riemann sums using \(N=10\) rectangles. Then, using the Fundamental Theorem of Calculus, Part 2 , determine the exact area. $$ \int(\cos x-\sin x) d x \text { over }[0, \pi] $$

In the following exercises, use a calculator to estimate the area under the curve by computing \(T_{10}\) , the average of the left- and right-endpoint Riemann sums using \(N=10\) rectangles. Then, using the Fundamental Theorem of Calculus, Part 2 , determine the exact area. $$ \int \frac{4}{x^{2}} d x \text { over }[1,4] $$

In the following exercises, evaluate each definite integral using the Fundamental Theorem of Calculus, Part 2 . $$ \int_{-1}^{2}\left(x^{2}-3 x\right) d x $$

In the following exercises, evaluate each definite integral using the Fundamental Theorem of Calculus, Part 2 . $$ \int_{-2}^{3}\left(x^{2}+3 x-5\right) d x $$

In the following exercises, evaluate each definite integral using the Fundamental Theorem of Calculus, Part 2 . $$ \int_{-2}^{3}(t+2)(t-3) d t $$

In the following exercises, evaluate each definite integral using the Fundamental Theorem of Calculus, Part 2 . $$ \int_{2}^{3}\left(t^{2}-9\right)\left(4-t^{2}\right) d t $$

In the following exercises, evaluate each definite integral using the Fundamental Theorem of Calculus, Part 2 . $$ \int_{1}^{2} x^{9} d x $$

In the following exercises, evaluate each definite integral using the Fundamental Theorem of Calculus, Part 2 . $$ \int_{0}^{1} x^{99} d x $$

In the following exercises, evaluate each definite integral using the Fundamental Theorem of Calculus, Part 2 . $$ \int_{4}^{8}\left(4 t^{5 / 2}-3 t^{3 / 2}\right) d t $$

In the following exercises, evaluate each definite integral using the Fundamental Theorem of Calculus, Part 2 . $$ \int_{1 / 4}^{4}\left(x^{2}-\frac{1}{x^{2}}\right) d x $$

In the following exercises, evaluate each definite integral using the Fundamental Theorem of Calculus, Part 2 . $$ \int_{1}^{2} \frac{2}{x^{3}} d x $$

In the following exercises, evaluate each definite integral using the Fundamental Theorem of Calculus, Part 2 . $$ \int_{1}^{4} \frac{1}{2 \sqrt{x}} d x $$

In the following exercises, evaluate each definite integral using the Fundamental Theorem of Calculus, Part 2 . $$ \int_{1}^{4} \frac{2-\sqrt{t}}{t^{2}} d t $$

In the following exercises, evaluate each definite integral using the Fundamental Theorem of Calculus, Part 2 . $$ \int_{1}^{16} \frac{d t}{t^{1 / 4}} $$

In the following exercises, evaluate each definite integral using the Fundamental Theorem of Calculus, Part 2 . $$ \int_{0}^{2 \pi} \cos \theta d \theta $$

In the following exercises, evaluate each definite integral using the Fundamental Theorem of Calculus, Part 2 . $$ \int_{0}^{\pi / 2} \sin \theta d \theta $$

In the following exercises, evaluate each definite integral using the Fundamental Theorem of Calculus, Part 2 . $$ \int_{0}^{\pi / 4} \sec ^{2} \theta d \theta $$

In the following exercises, evaluate each definite integral using the Fundamental Theorem of Calculus, Part 2 . \(\int_{0}^{\pi / 4} \sec \theta \tan \theta\)

In the following exercises, evaluate each definite integral using the Fundamental Theorem of Calculus, Part 2 . $$ \int_{\pi / 3}^{\pi / 4} \csc \theta \cot \theta d \theta $$

In the following exercises, evaluate each definite integral using the Fundamental Theorem of Calculus, Part 2 . $$ \int_{\pi / 4}^{\pi / 2} \csc ^{2} \theta d \theta $$

In the following exercises, evaluate each definite integral using the Fundamental Theorem of Calculus, Part 2 . $$ \int_{1}^{2}\left(\frac{1}{t^{2}}-\frac{1}{t^{3}}\right) d t $$

In the following exercises, evaluate each definite integral using the Fundamental Theorem of Calculus, Part 2 . $$ \int_{-2}^{-1}\left(\frac{1}{t^{2}}-\frac{1}{t^{3}}\right) d t $$

In the following exercises, use the evaluation theorem to express the integral as a function \(F(x)\) . $$ \int_{a}^{x} t^{2} d t $$

In the following exercises, use the evaluation theorem to express the integral as a function \(F(x)\) . $$ \int_{1}^{x} e^{t} d t $$

In the following exercises, use the evaluation theorem to express the integral as a function \(F(x)\) . $$ \int_{0}^{x} \cos t d t $$

In the following exercises, use the evaluation theorem to express the integral as a function \(F(x)\) . $$ \int_{-x}^{x} \sin t d t $$

In the following exercises, identify the roots of the integrand to remove absolute values, then evaluate using the Fundamental Theorem of Calculus, Part 2 . $$ \int_{-2}^{3}|x| d x $$

In the following exercises, identify the roots of the integrand to remove absolute values, then evaluate using the Fundamental Theorem of Calculus, Part 2 . $$ \int_{-2}^{4}\left|t^{2}-2 t-3\right| d t $$

In the following exercises, identify the roots of the integrand to remove absolute values, then evaluate using the Fundamental Theorem of Calculus, Part 2 . $$ \int_{0}^{\pi}|\cos t| d t $$

In the following exercises, identify the roots of the integrand to remove absolute values, then evaluate using the Fundamental Theorem of Calculus, Part 2 . $$ \int_{-\pi / 2}^{\pi / 2}|\sin t| d t $$

Suppose that the number of hours of daylight on a given day in Seattle is modeled by the function \(-3.75 \cos \left(\frac{\pi t}{6}\right)+12.25, \quad\) with \(t\) given in months and \(t=0\) corresponding to the winter solstice. a. What is the average number of daylight hours in a year? b. At which times \(t_{1} \quad\) and \(\quad t_{2}, \quad\) where \(0 \leq t_{1} < t_{2} < 12,\) do the number of daylight hours equal the average number? c. Write an integral that expresses the total number of daylight hours in Seattle between \(t_{1}\) and \(t_{2}\) . d. Compute the mean hours of daylight in Seattle between \(t_{1}\) and \(t_{2}, \quad\) where \(0 \leq t_{1} < t_{2} < 12,\) and then between \(t_{2}\) and \(t_{1},\) and show that the average of the two is equal to the average day length.

Suppose the rate of gasoline consumption in the United States can be modeled by a sinusoidal function of the form \(\left(11.21-\cos \left(\frac{\pi t}{6}\right)\right) \times 10^{9} \mathrm{gal} / \mathrm{mo}\). a. What is the average monthly consumption, and for which values of \(t\) is the rate at time \(t\) equal to the average rate? b. What is the number of gallons of gasoline consumed in the United States in a year? c. Write an integral that expresses the average monthly U.S. gas consumption during the part of the year between the beginning of April \((t=3)\) and the end of September \((t=9)\).

Explain why, if \(f\) is continuous over \([a, b],\) there is at least one point \(c \in[a, b] \quad\) such \(\quad\) that. $$f(c)=\frac{1}{b-a} \int_{a}^{b} f(t) d t$$

Explain why, if \(f\) is continuous over \([a, b]\) and is not equal to a constant, there is at least one point \(M \in[a, b]\) such that \(f(M)=\frac{1}{b-a} \int_{a}^{b} f(t) d t\) and at least one point \(m \in[a, b]\) such that \(f(m) < \frac{1}{b-a} \int_{a}^{b} f(t) d t\)

Kepler’s first law states that the planets move in elliptical orbits with the Sun at one focus. The closest point of a planetary orbit to the Sun is called the perihelion (for Earth, it currently occurs around January 3) and the farthest point is called the aphelion (for Earth, it currently occurs around July 4). Kepler’s second law states that planets sweep out equal areas of their elliptical orbits in equal times. Thus, the two arcs indicated in the following figure are swept out in equal times. At what time of year is Earth moving fastest in its orbit? When is it moving slowest?

A point on an ellipse with major axis length 2\(a\) and minor axis length 2\(b\) has the coordinates \((a \cos \theta, b \sin \theta), 0 \leq \theta \leq 2 \pi\) a. Show that the distance from this point to the focus at \((-c, 0) \quad\) is \(\quad d(\theta)=a+c \cos \theta, \quad\) where \(c=\sqrt{a^{2}-b^{2}}\) b. Use these coordinates to show that the average distance \(\overline{d}\) from a point on the ellipse to the focus at \((-c, 0),\) with respect to angle \(\theta,\) is \(a .\)

A point on an ellipse with major axis length \(2 a\) and minor axis length \(2 b\) has the coordinates \((a \cos \theta, b \sin \theta), 0 \leq \theta \leq 2 \pi\) a. Show that the distance from this point to the focus at \((-c, 0)\) is \(d(\theta)=a+c \cos \theta, \quad\) where \(c=\sqrt{a^{2}-b^{2}}\) b. Use these coordinates to show that the average distance \(\bar{d}\) from a point on the ellipse to the focus at \((-c, 0),\) with respect to angle \(\theta\), is \(a\).

As implied earlier, according to Kepler's laws, Earth's orbit is an ellipse with the Sun at one focus. The perihelion for Earth's orbit around the Sun is \(147,098,290\) \(\mathrm{km}\) and the aphelion is \(152,098,232 \mathrm{km} .\) a. By placing the major axis along the \(x\) -axis, find the average distance from Earth to the Sun. b. The classic definition of an astronomical unit (AU) is the distance from Earth to the Sun, and its value was computed as the average of the perihelion and aphelion distances. Is this definition justified?

The force of gravitational attraction between the Sun and a planet is \(F(\theta)=\frac{G m M}{r^{2}(\theta)},\) where \(m\) is the mass of the planet, \(M\) is the mass of the Sun, \(G\) is a universal constant, and \(r(\theta)\) is the distance between the Sun and the planet when the planet is at an angle \(\theta\) with the major axis of its orbit. Assuming that \(M, m,\) and the ellipse parameters \(a\) and \(b\) (half-lengths of the major and minor axes) are given, set up-but do not evaluate - an integral that expresses in terms of \(G, m, M, a, b\) the average gravitational force between the Sun and the planet.

The displacement from rest of a mass attached to a spring satisfies the simple harmonic motion equation \(x(t)=A \cos (\omega t-\phi),\) where \(\phi\) is a phase constant, \(\omega\) is the angular frequency, and \(A\) is the amplitude. Find the average velocity, the average speed (magnitude of velocity), the average displacement, and the average distance from rest (magnitude of displacement) of the mass.

Use basic integration formulas to compute the following antiderivatives. $$ \int\left(\sqrt{x}-\frac{1}{\sqrt{x}}\right) d x $$

Use basic integration formulas to compute the following antiderivatives. $$ \int\left(e^{2 x}-\frac{1}{2} e^{x / 2}\right) d x $$

Use basic integration formulas to compute the following antiderivatives. $$ \int \frac{d x}{2 x} $$

Use basic integration formulas to compute the following antiderivatives. $$ \int \frac{x-1}{x^{2}} d x $$

Use basic integration formulas to compute the following antiderivatives. $$ \int_{0}^{\pi}(\sin x-\cos x) d x $$

Use basic integration formulas to compute the following antiderivatives. $$ \int_{0}^{\pi / 2}(x-\sin x) d x $$

Write an integral that expresses the increase in the perimeter \(P(s)\) of a square when its side length \(s\) increases from 2 units to 4 units and evaluate the integral.

Write an integral that quantifies the change in the area \(A(s)=s^{2}\) of a square when the side length doubles from \(S\) units to 2 \(\mathrm{S}\) units and evaluate the integral.