## 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 $\displaystyle{\int x^2 \sqrt{x-1}\, dx.}$
1. September 7, 2011 at 6:39 pm

Are you going to post the answers for this before the test tomorrow?

2. December 5, 2011 at 5:56 pm

For number 7, I get arcsin (x/k) = pi*2k. Now, where do I go from there? I know that the x value is anywhere from -k to k, but in any case I will just get an angle. How do I find something that will produce the 2k part of the answer?

• December 5, 2011 at 9:00 pm

Solve that equation for the corresponding value of $x$, that’s all.

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