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) ${\displaystyle \int x^{2}{\sqrt {1+x^{3}}}~dx}$
b) ${\displaystyle \int _{\frac {\pi }{4}}^{\frac {\pi }{2}}{\frac {\cos(x)}{\sin ^{2}(x)}}~dx}$

Problem 2

Find the average value of the function on the given interval.

${\displaystyle f(x)=2x^{3}(1+x^{2})^{4},~~~[0,2]}$

Problem 3

Evaluate the indefinite and definite integrals.

a) ${\displaystyle \int x^{2}e^{x}~dx}$
b) ${\displaystyle \int _{1}^{e}x^{3}\ln x~dx}$

Problem 4

Evaluate the integral:

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

Problem 5

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

a) Compute the left-hand Riemann sum approximation of ${\displaystyle \int _{0}^{3}f(x)~dx}$ with ${\displaystyle n=3}$ boxes.
b) Compute the right-hand Riemann sum approximation of ${\displaystyle \int _{0}^{3}f(x)~dx}$ with ${\displaystyle n=3}$ boxes.
c) Express ${\displaystyle \int _{0}^{3}f(x)~dx}$ as a limit of right-hand Riemann sums (as in the definition of the definite integral). Do not evaluate the limit.