1. (a) Find the area of the region enclosed by the curves \( y=1-2 x^{2} \) and \( y=4 x^{2}-1 \). (b) Find the average value \( f_{\text {ave }} \) of the function \( f(x)=\ln x \) on the interval \( [1, e] \). 2. (a) Find volume of rutating region bounded by \( y=x^{5}, y=x, x=0, x=1 \), about the \( x \)-axis. (b) Find the resulting volume when the region between \( y=e^{-x^{2}} \) and \( y=0 \) and to the right of \( x=0 \) is rotated about the \( y \)-axis. 3. A circular swimming pool has a diameter of \( 12 \mathrm{ft} \), the sides are \( 3 \mathrm{ft} \) high, and the water is \( 2 \mathrm{ft} \) deep. Given that water weighs \( 62.5 \mathrm{lb} / \mathrm{ft}^{3} \), do the following. (a) Express the work needed to pump all of the water over the side as a limit of a Riemann sum. (b) Evaluate the limit in part (a) by expressing it as an integral.
Solution
Step 1 ：\( \text{(a) Intersection points: } x = \pm \frac{1}{\sqrt{3}} \)
Step 2 ：\( \text{Area} = \int_{-1/\sqrt{3}}^{1/\sqrt{3}} ((4x^2 - 1) - (1 - 2x^2)) \, dx \)
Step 3 ：\( \text{Area} = \frac{4}{3} \)
Step 4 ：\( \text{(b)} f_{ave} = \frac{1}{e-1} \int_{1}^{e} \ln{x} \, dx \)
Step 5 ：\( \text{Average Value} = \frac{1}{2} \)
Step 6 ：\( \text{2(a) Volume} = \pi \int_{0}^{1} (x^5)^2 - (x)^2 \, dx \)
Step 7 ：\( \text{Volume} = \frac{\pi}{21} \)
Step 8 ：\( \text{2(b)} x = \left[-\frac{1}{2}\ln{y}\right] \)
Step 9 ：\( \text{Volume} = 2\pi \int_{0}^{1} y[-\frac{1}{2}\ln{y} ] \, dy \)
Step 10 ：\( \text{Volume} = \frac{\pi}{4} \)
Step 11 ：\( \text{3(a)} W = \lim_{n\to\infty}\sum_{i=1}^{n} 2\pi xi^{\prime} (1 - \frac{i^{\prime 2}}{144})(3 - \frac{i^{\prime}}{6}) \Delta x \)
Step 12 ：\( \text{3(b)} W = 2\pi \cdot 62.5 \int_{0}^{6} x(3 - \sqrt{144 - x^2}) \, dx \)
Step 13 ：\( \text{Work} = 4500\pi \)
