Math 2 - Review Problems for Exam 2

1. Evaluate ${\displaystyle \int_0^3 x e^{-x}\ dx}$. Answer: $-4 e^{-3}+1$

2. Evaluate ${\displaystyle \int_e^{e^2} \sqrt{x} \ln x \ dx}$. Answer: $\frac{1}{9}(8 e^3 - 2 e^{3/2})$

3. Evaluate ${\displaystyle \int_0^3 t(t^2+16)^{1/2} \ dx}$. Answer: $\frac{61}{3}$

4. Evaluate ${\displaystyle \int_0^{(e-1)/2} \frac{\ln (2x+1)^4}{2x+1} \ dx}$. Answer: $1$

5. Evaluate ${\displaystyle \int_0^{e^3} f(x) \ dx}$ where
   $$f(x)= \left\{ \begin{array}{ll} 2e^x & 0 \leq x \leq 1\\ 2x-2 & 1<x<e \\ (\ln x)^2/x & e \leq x \leq e^3 \end{array} \right.$$
   Answer: $e^2 + \frac{23}{3}$

6. Evalute ${\displaystyle \int_1^e (1+\ln x) \ dx}$. Answer: $e$

7. Use the limit process to evaluate ${\displaystyle \int_0^2 (4x+3) \ dx}$. Then think of two ways to check your answer.

   Answer: $14$

8. Use the limit process to evaluate ${\displaystyle \int_0^3 (x^2+1) \ dx}$.

   Answer: $12$

9. Find the area of the region bounded by the curve ${\displaystyle y = 3-x^2}$ and the lines $y=4-x$, $x=-1$, and $x=2$.

   Answer: $4.5$

10. Find the area of the region bounded by ${\displaystyle x=y^2-y}$ and $x=y$.

    Answer: $\frac{4}{3}$

11. Consider the region bounded by ${\displaystyle y=x^2}$ and $y=4$. Find the volume of the solid generated when R is revolved about (a) the $x$-axis, (b) the $y$-axis.

    Answer: (a) $\frac{256 \pi}{5}$, (b) $8 \pi$

12. Consider the region bounded by ${\displaystyle y=\sqrt x}$, $y=0$ and $x=4$. Find the volume of the solid generated when R is revolved about (a) the $x$-axis, (b) the $y$-axis.

    Answer: (a) $8 \pi$, (b) $25.6 \pi$