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Calculus II -- Fall 2012 -- midterm

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  • Write the problems in the given order, each problem on a separate page.
  • Show enough work to justify your answers. Quote theorems and formulas in the book when necessary.
  • Don't simplify unless very easy or absolutely necessary.

1. Find the volume of a right circular cone of radius $R$ and height $h$ by any method you like.

2. Compute the average area of the cross section of the sphere of radius 1.

3. Integrate by parts: $$\int x(\ln x)^2 dx.$$

4. Use substitution to evaluate the integral: $$\int _0^1 \frac{1}{\sqrt{4-x^2}} dx.$$

5. Use the table of integrals to evaluate: $$\int x^2(\sqrt{x^2-4}-\sqrt{x^2+9})dx.$$

6. Write the mid-point Riemann sum that approximates the integral $\int _0^1 \sin x dx$ within $.01$.

7. Find the center of mass of the region below $y=2x$ for $0 \leq x \leq 1$.

8. Find the Taylor polynomial of degree $4$ that would help to approximate $e^{1.01}$.