1. (a) Find the area of the region enclosed by the curves $y=1-2x^2$ and $y=4x^2-1$.
(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.

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 ：$\text{(b)}{f}_{ave}=\frac{1}{e-1}{\int }_1^e\ln xdx$

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 ：$\text{2(b)}x=[-\frac{1}{2}\ln y]$

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 ：$\text{3(a)}W=\underset{n\to \mathrm{\infty }}{lim}\sum _{i=1}^{n}2\pi x{i}^{\prime }(1-\frac{i^{\prime 2}}{144})(3-\frac{i^{\prime }}{6})\mathrm{\Delta }x$

Step 12 ：$\text{3(b)}W=2\pi \cdot 62.5{\int }_0^{6}x(3-\sqrt{144-x^2})dx$

Step 13 ：$\text{Work}=4500\pi$