007B Sample Final 2

This is a sample, and is meant to represent the material usually covered in Math 7B 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 

Consider the area bounded by the following two functions:

$y=\cos x{\text{ and }}y=2-\cos x,~0\leq x\leq 2\pi .$

(a) Sketch the graphs and find their points of intersection.

(b) Find the area bounded by the two functions.

 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 

A city bordered on one side by a lake can be approximated by a semicircle of radius 7 miles, whose city center is on the shoreline. As we move away from the center along a radius the population density of the city can be approximated by:

$\rho (x)=25000e^{-0.15x}$

people per square mile. What is the population of the city?

 Problem 5 

(a) Find the area of the surface obtained by rotating the arc of the curve

$y^{3}=x$

between  $(0,0)$  and  $(1,1)$  about the  $y$-axis.

(b) Find the length of the arc

$y=1+9x^{\frac {3}{2}}$

between the points  $(1,10)$  and  $(4,73).$

 Problem 6 

Evaluate the following integrals:

(a)  $\int {\frac {dx}{x^{2}{\sqrt {x^{2}-16}}}}$

(b)  $\int _{-\pi }^{\pi }\sin ^{3}x\cos ^{3}x~dx$

(c)  $\int _{0}^{1}{\frac {x-3}{x^{2}+6x+5}}~dx$

 Problem 7 

Evaluate the following integrals or show that they are divergent:

(a)  $\int _{1}^{\infty }{\frac {\ln x}{x^{4}}}~dx$

(b)  $\int _{0}^{1}{\frac {3\ln x}{\sqrt {x}}}~dx$