009B Sample Final 2

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 

a) State both parts of the Fundamental Theorem of Calculus.

b) Evaluate the integral

$\int _{0}^{1}{\frac {d}{dx}}{\bigg (}e^{\tan ^{-1}(x)}{\bigg )}dx$

c) Compute

${\frac {d}{dx}}\int _{1}^{\frac {1}{x}}\sin t~dt$

 Problem 2 

Find the area of the region between the two curves $y=3x-x^{2}$ and $y=2x^{3}-x^{2}-5x.$

 Problem 3 

Find the volume of the solid obtained by rotating the region bounded by the curves $y=x$ and $y=x^{2}$ about the line $y=2.$

 Problem 4 

Compute the following integrals.

a) $\int e^{x}(x+\sin(e^{x}))~dx$

b) $\int {\frac {2x^{2}+1}{2x^{2}+x}}~dx$

c) $\int \sin ^{3}x~dx$

 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.