# Difference between revisions of "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

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.

## Problem 2

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 3

A population grows at a rate

${\displaystyle P'(t)=500e^{-t}}$

where  ${\displaystyle P(t)}$  is the population after  ${\displaystyle t}$  months.

(a)   Find a formula for the population size after  ${\displaystyle t}$  months, given that the population is  ${\displaystyle 2000}$  at  ${\displaystyle t=0.}$

(b)   Use your answer to part (a) to find the size of the population after one month.

## Problem 4

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 5

Evaluate the integral:

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