007B Sample Final 3

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 

Consider the region bounded by the following two function:

$y=-x^{2}+9{\text{ and }}y=0.$

(a) Using the Left Hand Rule with three rectangles having equal width, approximate the area.

(b) Using the Right Hand Rule with three rectangles having equal width, approximate the area.

 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:

$y=\sin x{\text{ and }}y={\frac {2}{\pi }}x.$

 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 

Find the volume of the solid obtained by rotating the region bounded by the following about  $x$-axis.

$y={\sqrt {\sin x}},~0\leq x\leq \pi ,~y=0$

 Problem 6 

Evaluate the improper integrals:

(a)  $\int _{0}^{\infty }xe^{-x}~dx$

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

 Problem 7 

In a single-compartment model, the concentration of the solution at time  $t,$  denoted by  $C(t),$  is described by the differential equation:

${\frac {dC}{dt}}={\frac {0.2}{400}}(3-C).$

Solve the above differential equation when  $C(0)=0,$  and find  $\displaystyle {\lim _{t\rightarrow \infty }C(t)}.$