Chapter 12

Use the table of integrals to find the exact area of the region bounded by the graphs of the equations. Then use a graphing utility to graph the region and approximate the area. $$ y=\frac{x}{\sqrt{x+1}}, y=0, x=8 $$

$$ \text { Evaluate the definite integral. } $$ $$ \int_{0}^{1} \frac{x^{3}}{x^{2}-2} d x $$

Find the indefinite integral. (Hint: Integration by parts is not required for all the integrals.) $$ \int \frac{x e^{2 x}}{(2 x+1)^{2}} d x $$

Use a spreadsheet to complete the table for the specified values of \(a\) and \(n\) to demonstrate that \(\lim _{x \rightarrow \infty} x^{n} e^{-a x}=0, \quad a>0, n>0\) \begin{tabular}{|l|l|l|l|l|} \hline\(x\) & 1 & 10 & 25 & 50 \\ \hline\(x^{n} e^{-a x}\) & & & & \\ \hline \end{tabular} $$ a=\frac{1}{2}, n=5 $$

Use the error formulas to find bounds for the error in approximating the integral using (a) the Trapezoidal Rule and (b) Simpson's Rule. (Let \(n=4 .\).) $$ \int_{0}^{1} e^{-x^{2}} d x $$

Use the table of integrals to find the exact area of the region bounded by the graphs of the equations. Then use a graphing utility to graph the region and approximate the area. $$ y=\frac{2}{1+e^{4 x}}, y=0, x=0, x=1 $$

$$ \text { Evaluate the definite integral. } $$ $$ \int_{0}^{1} \frac{x^{3}-1}{x^{2}-4} d x $$

Find the indefinite integral. (Hint: Integration by parts is not required for all the integrals.) $$ \int \frac{x^{3} e^{x^{2}}}{\left(x^{2}+1\right)^{2}} d x $$

Use the error formulas to find \(n\) such that the error in the approximation of the definite integral is less than \(0.0001\) using (a) the Trapezoidal Rule and (b) Simpson's Rule. $$ \int_{0}^{1} x^{3} d x $$

Use the table of integrals to find the exact area of the region bounded by the graphs of the equations. Then use a graphing utility to graph the region and approximate the area. $$ y=\frac{x}{1+e^{x^{2}}}, y=0, x=2 $$

$$ \text { Evaluate the definite integral. } $$$$ \int_{1}^{2} \frac{x^{3}-4 x^{2}-3 x+3}{x^{2}-3 x} d x $$

Evaluate the definite integral. $$ \int_{1}^{2} x^{2} e^{x} d x $$

Use the error formulas to find \(n\) such that the error in the approximation of the definite integral is less than \(0.0001\) using (a) the Trapezoidal Rule and (b) Simpson's Rule. $$ \int_{1}^{3} \frac{1}{x} d x $$

Use the table of integrals to find the exact area of the region bounded by the graphs of the equations. Then use a graphing utility to graph the region and approximate the area. $$ y=\frac{-e^{x}}{1-e^{2 x}}, y=0, x=1, x=2 $$

$$ \text { Evaluate the definite integral. } $$ $$ \int_{2}^{4} \frac{x^{4}-4}{x^{2}-1} d x $$

Evaluate the definite integral. $$ \int_{0}^{2} \frac{x^{2}}{e^{x}} d x $$

Use the table of integrals to find the exact area of the region bounded by the graphs of the equations. Then use a graphing utility to graph the region and approximate the area. $$ y=x^{2} \sqrt{x^{2}+4}, y=0, x=\sqrt{5} $$

Evaluate the definite integral. $$ \int_{0}^{4} \frac{x}{e^{x / 2}} d x $$

Evaluate the definite integral. $$ \int_{1}^{2} x^{2} \ln x d x $$

Women's Height The mean height of American women between the ages of 30 and 39 is \(64.5\) inches, and the standard deviation is \(2.7\) inches. Find the probability that a 30 - to 39 -year-old woman chosen at random is (a) between 5 and 6 feet tall. (b) 5 feet 8 inches or taller. (c) 6 feet or taller

Use a program similar to the Simpson's Rule program on page 906 to approximate the integral. Use \(n=100\). $$ \int_{1}^{4} x \sqrt{x+4} d x $$

Evaluate the definite integral. $$ \int_{0}^{1} \frac{x}{\sqrt{1+x}} d x $$

Evaluate the definite integral. $$ \int_{1}^{e} x^{5} \ln x d x $$

Quality Control A company manufactures wooden yardsticks. The lengths of the yardsticks are normally distributed with a mean of 36 inches and a standard deviation of \(0.2\) inch. Find the probability that a yardstick is (a) longer than \(35.5\) inches. (b) longer than \(35.9\) inches.

Use a program similar to the Simpson's Rule program on page 906 to approximate the integral. Use \(n=100\). $$ \int_{1}^{4} x^{2} \sqrt{x+4} d x $$

Evaluate the definite integral. $$ \int_{0}^{5} \frac{x}{\sqrt{5+2 x}} d x $$

Evaluate the definite integral. $$ \int_{1}^{e} 2 x \ln x d x $$

Determine the amount of money required to set up a charitable endowment that pays the amount \(P\) each year indefinitely for the annual interest rate \(r\) compounded continuously. $$ P=\$ 5000, r=7.5 \% $$

Use a program similar to the Simpson's Rule program on page 906 to approximate the integral. Use \(n=100\). $$ \int_{2}^{5} 10 x e^{-x} d x $$

Evaluate the definite integral. $$ \int_{0}^{5} \frac{x}{(4+x)^{2}} d x $$

Find the area of the region bounded by the graphs of the given equations. $$ y=\frac{12}{x^{2}+5 x+6}, y=0, x=0, x=1 $$

Evaluate the definite integral. $$ \int_{-1}^{0} \ln (x+2) d x $$

Determine the amount of money required to set up a charitable endowment that pays the amount \(P\) each year indefinitely for the annual interest rate \(r\) compounded continuously. $$ P=\$ 12,000, r=6 \% $$

Use a program similar to the Simpson's Rule program on page 906 to approximate the integral. Use \(n=100\). $$ \int_{2}^{5} 10 x^{2} e^{-x} d x $$

Evaluate the definite integral. $$ \int_{2}^{4} \frac{x^{2}}{(3 x-5)} d x $$

Find the area of the region bounded by the graphs of the given equations. $$ y=\frac{-24}{x^{2}-16}, y=0, x=1, x=3 $$

Evaluate the definite integral. $$ \int_{0}^{1} \ln (1+2 x) d x $$

MAKE A DECISION: SCHOLARSHIP FUND You want to start a scholarship fund at your alma mater. You plan to give one \(\$ 18,000\) scholarship annually beginning one year from now and you have at most \(\$ 400,000\) to start the fund. You also want the scholarship to be given out indefinitely. Assuming an annual interest rate of \(5 \%\) compounded continuously, do you have enough money for the scholarship fund?

Evaluate the definite integral. $$ \int_{0}^{4} \frac{6}{1+e^{0.5 x}} d x $$

Write the partial fraction decomposition for the rational expression. Check your result algebraically. Then assign a value to the constant \(a\) and use a graphing utility to check the result graphically. $$ \frac{1}{a^{2}-x^{2}} $$

Find the area of the region bounded by the graphs of the equations. Then use a graphing utility to graph the region and verify your answer. $$ y=x^{3} e^{x}, y=0, x=0, x=2 $$

MAKE A DECISION: CHARITABLE FOUNDATION A charitable foundation wants to help schools buy computers. The foundation plans to donate \(\$ 35,000\) each year to one school beginning one year from now, and the foundation has at most \(\$ 500,000\) to start the fund. The foundation wants the donation to be given out indefinitely. Assuming an annual interest rate of \(8 \%\) compounded continuously, does the foundation have enough money to fund the donation?

Evaluate the definite integral. $$ \int_{2}^{4} \sqrt{3+x^{2}} d x $$

Write the partial fraction decomposition for the rational expression. Check your result algebraically. Then assign a value to the constant \(a\) and use a graphing utility to check the result graphically. $$ \frac{1}{x(x+a)} $$

Find the area of the region bounded by the graphs of the equations. Then use a graphing utility to graph the region and verify your answer. $$ y=\left(x^{2}-1\right) e^{x}, y=0, x=-1, x=1 $$

Present Value A business is expected to yield a continuous flow of profit at the rate of \(\$ 500,000\) per year. If money will earn interest at the nominal rate of \(9 \%\) per year compounded continuously, what is the present value of the business (a) for 20 years and (b) forever?

Evaluate the definite integral. $$ \int_{1}^{4} x \ln x d x $$

Find the area of the region bounded by the graphs of the equations. Then use a graphing utility to graph the region and verify your answer. $$ y=x^{2} \ln x, y=0, x=1, x=e $$

Arc Length A fleeing hare leaves its burrow \((0,0)\) and moves due north (up the \(y\) -axis). At the same time, a pursuing lynx leaves from 1 yard east of the burrow \((1,0)\) and always moves toward the fleeing hare (see figure). If the lynx's speed is twice that of the hare's, the equation of the lynx's path is \(y=\frac{1}{3}\left(x^{3 / 2}-3 x^{1 / 2}+2\right)\) Find the distance traveled by the lynx by integrating over the interval \([0,1]\).

Use the definite integral below to find the required arc length. If \(f\) has a continuous derivative, then the arc length of \(f\) between the points \((a, f(a))\) and \((b, f(b))\) is \(\int_{b}^{a} \sqrt{1+\left[f^{\prime}(x)\right]^{2}} d x\) Arc Length A fleeing hare leaves its burrow \((0,0)\) and moves due north (up the \(y\) -axis). At the same time, a pursuing lynx leaves from 1 yard east of the burrow \((1,0)\) and always moves toward the fleeing hare (see figure). If the lynx's speed is twice that of the hare's, the equation of the lynx's path is \(y=\frac{1}{3}\left(x^{3 / 2}-3 x^{1 / 2}+2\right)\) Find the distance traveled by the lynx by integrating over the interval \([0,1]\).