009B Sample Final 3

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

Click on the  boxed problem numbers  to go to a solution.

 Problem 1 

Divide the interval $[-1,1]$ into four subintervals of equal length ${\frac {1}{2}}$ and compute the left-endpoint Riemann sum of $y=1-x^{2}.$

 Problem 2 

Evaluate the following integrals.

a) $\int _{0}^{\frac {\sqrt {3}}{4}}{\frac {1}{1+16x^{2}}}~dx$

b) $\int {\frac {x^{2}}{(1+x^{3})^{2}}}$

c) $\int _{1}^{e}{\frac {\cos(\ln(x))}{x}}~dx$

 Problem 3 

The population density of trout in a stream is

$\rho (x)=|-x^{2}+6x+16|$

where $\rho$ is measured in trout per mile and $x$ is measured in miles. $x$ runs from 0 to 12.

a) Graph $\rho (x)$ and find the minimum and maximum.

b) Find the total number of trout in the stream.

 Problem 4 

Find the volume of the solid obtained by rotating about the $x$-axis the region bounded by $y={\sqrt {1-x^{2}}}$ and $y=0.$

 Problem 5 

Consider the solid obtained by rotating the area bounded by the following three functions about the $y$-axis:

$x=0$, $y=e^{x}$, and $y=ex$.

a) Sketch the region bounded by the given three functions. Find the intersection point of the two functions:

$y=e^{x}$ and $y=ex$. (There is only one.)

b) Set up the integral for the volume of the solid.

c) Find the volume of the solid by computing the integral.

 Problem 6 

Evaluate the improper integrals:

a) $\int _{0}^{\infty }xe^{-x}~dx$

b) $\int _{1}^{4}{\frac {dx}{\sqrt {4-x}}}$

 Problem 7 

a) Find the length of the curve

$y=\ln(\cos x),~~~0\leq x\leq {\frac {\pi }{3}}$.

b) The curve

$y=1-x^{2},~~~0\leq x\leq 1$

is rotated about the $y$-axis. Find the area of the resulting surface.