## First midterm-Practice test

This is a practice exam for the first midterm. Feel free to drop any comment or question below. I will try to answer here as many as possible, until Wednesday Sept 07. Also, remember that Alex is holding a review session during his recitation on Monday Sept 05, and Kevin will be available during his usual SI sessions. Good luck!

1. [10 pts] Find the area of the region that is enclosed between the curves $y=x^2$ and $y=x+6.$

2. [15 pts] Find the volume of the solid that is obtained when the region under the curve $y=\sqrt{x}$ over the interval $[1,4]$ is revolved about the $x$-axis.

3. [15 pts] Find the volume of the solid generated when the region enclosed by $y=\sqrt{x}, y=2$ and $x=0$ is revolved about the $y$-axis.

4. [10 pts] Use the Fundamental Theorem of Calculus to find the derivative of the function $F(x)=\displaystyle{\int_x^\pi \sqrt{1+\sec t}\, dt.}$

5. [15 pts] Find the volume of the solid obtained by rotating the region bounded by the curves $x=4y^2-y^3$ and $x=0$ about the $x$–axis.

6. [10 pts] Find the average value of the function $f(x) = 1/x$ over the interval $[1,e].$

7. [15 pts] Find a positive value of $k$ such that the average value of $f(x) = \displaystyle{\frac{1}{\sqrt{k^2-x^2}}}$ over the interval $[-k,k]$ is $\pi$.

8. [10 pts] Evaluate the integral $\displaystyle{\int x^2 \sqrt{x-1}\, dx.}$
Solve that equation for the corresponding value of $x$, that’s all.