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!

[Math Cheat Sheet T-shirt] [Engineer Cheat Sheet T-shirt]
  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. Spencer Swartzel
    September 7, 2011 at 6:39 pm

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

  2. Anonymous
    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?

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