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Chapter 12: Problem 46
Find the area of the region bounded by the graphs of the given equations. $$ y=\frac{-24}{x^{2}-16}, y=0, x=1, x=3 $$
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Capitalized Cost In Exercises 51 and 52, find the capitalized cost \(C\) of an asset \((a)\) for \(n=5\) years, \((b)\) for \(n=10\) years, and (c) forever. The capitalized cost is given by \(C=C_{0}+\int_{0}^{n} c(t) e^{-r t} d t\) where \(C_{0}\) is the original investment, \(t\) is the time in years, \(r\) is the annual interest rate compounded continuously, and \(c(t)\) is the annual cost of maintenance (measured in dollars). [Hint: For part (c), see Exercises \(35-38 .]\) $$ C_{0}=\$ 650,000, c(t)=25,000, r=10 \% $$
Capitalized Cost In Exercises 51 and 52, find the capitalized cost \(C\) of an asset \((a)\) for \(n=5\) years, \((b)\) for \(n=10\) years, and (c) forever. The capitalized cost is given by \(C=C_{0}+\int_{0}^{n} c(t) e^{-r t} d t\) where \(C_{0}\) is the original investment, \(t\) is the time in years, \(r\) is the annual interest rate compounded continuously, and \(c(t)\) is the annual cost of maintenance (measured in dollars). [Hint: For part (c), see Exercises \(35-38 .]\) $$ C_{0}=\$ 650,000, c(t)=25,000(1+0.08 t), r=12 \% $$
Use the Trapezoidal Rule and Simpson's Rule to approximate the value of the definite integral for the indicated value of \(n\). Compare these results with the exact value of the definite integral. Round your answers to four decimal places. $$ \int_{1}^{2} \frac{1}{x} d x, n=4 $$
Determine whether the improper integral diverges or converges. Evaluate the integral if it converges. $$ \int_{1}^{\infty} \frac{e^{\sqrt{x}}}{\sqrt{x}} d x $$
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 \([0,1]\).
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