009B Sample Final 1

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 

Consider the region bounded by the following two functions:

$y=2(-x^{2}+9)$  and  $y=0$.

(a) Using the lower sum with three rectangles having equal width, approximate the area.

(b) Using the upper sum with three rectangles having equal width, approximate the area.

(c) Find the actual area of the region.

 Problem 2 

We would like to evaluate

${\frac {d}{dx}}{\bigg (}\int _{-1}^{x}\sin(t^{2})2t\,dt{\bigg )}.$

(a) Compute  $f(x)=\int _{-1}^{x}\sin(t^{2})2t\,dt$.

(b) Find  $f'(x)$.

(c) State the Fundamental Theorem of Calculus.

(d) Use the Fundamental Theorem of Calculus to compute  ${\frac {d}{dx}}{\bigg (}\int _{-1}^{x}\sin(t^{2})2t\,dt{\bigg )}$  without first computing the integral.

 Problem 3 

Consider the area bounded by the following two functions:

$y=\cos x$  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 4 

Compute the following integrals.

(a)  $\int {\frac {t^{2}}{\sqrt {1-t^{6}}}}~dt$

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

(c)  $\int \sin ^{3}x~dx$

 Problem 5 

The region bounded by the parabola  $y=x^{2}$  and the line  $y=2x$  in the first quadrant is revolved about the  $y$-axis to generate a solid.

(a) Sketch the region bounded by the given functions and find their points of intersection.

(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.