009B Sample Midterm 2

This is a sample, and is meant to represent the material usually covered in Math 9B for the midterm. An actual test may or may not be similar. Click on the

 boxed problem numbers  to go to a solution.

 Problem 1 

Consider the region $S$ bounded by $x=1,x=5,y={\frac {1}{x^{2}}}$ and the $x$-axis.

a) Use four rectangles and a Riemann sum to approximate the area of the region $S$. Sketch the region $S$ and the rectangles and indicate whether your rectangles overestimate or underestimate the area of $S$.

b) Find an expression for the area of the region $S$ as a limit. Do not evaluate the limit.

 Problem 2 

This problem has three parts:

a) State the fundamental theorem of calculus.

b) Compute ${\frac {d}{dx}}\int _{0}^{\cos(x)}\sin(t)~dt$

c) Evaluate $\int _{0}^{\frac {\pi }{4}}\sec ^{2}x~dx$

 Problem 3 

Evaluate

a) $\int _{1}^{2}{\bigg (}2t+{\frac {3}{t^{2}}}{\bigg )}{\bigg (}4t^{2}-{\frac {5}{t}}{\bigg )}~dt$

b) $\int _{0}^{2}(x^{3}+x){\sqrt {x^{4}+2x^{2}+4}}~dx$

 Problem 4 

Evaluate the integral:

$\int e^{-2x}\sin(2x)~dx$

 Problem 5 

Evaluate the integral:

$\int \tan ^{4}x~dx$