009B Sample Midterm 1

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 

Evaluate the indefinite and definite integrals.

(a)   $\int x^{2}{\sqrt {1+x^{3}}}~dx$

(b)   $\int _{\frac {\pi }{4}}^{\frac {\pi }{2}}{\frac {\cos(x)}{\sin ^{2}(x)}}~dx$

 Problem 2 

Otis Taylor plots the price per share of a stock that he owns as a function of time

and finds that it can be approximated by the function

$s(t)=t(25-5t)+18$

where $t$ is the time (in years) since the stock was purchased.

Find the average price of the stock over the first five years.

 Problem 3 

Evaluate the indefinite and definite integrals.

(a)   $\int x^{2}e^{x}~dx$

(b)   $\int _{1}^{e}x^{3}\ln x~dx$

 Problem 4 

Evaluate the integral:

$\int \sin ^{3}x\cos ^{2}x~dx$

 Problem 5 

Let $f(x)=1-x^{2}$.

(a) Compute the left-hand Riemann sum approximation of $\int _{0}^{3}f(x)~dx$ with $n=3$ boxes.

(b) Compute the right-hand Riemann sum approximation of $\int _{0}^{3}f(x)~dx$ with $n=3$ boxes.

(c) Express $\int _{0}^{3}f(x)~dx$ as a limit of right-hand Riemann sums (as in the definition of the definite integral). Do not evaluate the limit.