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Problem  1   2   3   4  Total
Score          
MAT 126
Calculus B

Midterm 2

April 5, 2000

SHOW ALL YOUR WORK ON THESE PAGES! TOTAL SCORE = 100



1.
(30 points) Evaluate each of the following definite integrals.

(a)
${\displaystyle \int_0^1 \sin ( \pi x)~dx}$

(b)
${\displaystyle \int_0^1 x \sqrt{1 - x^2}~dx}$

(c)
${\displaystyle \int_{-1}^1 x e^x ~dx}$

2.
(30 points) Find the following antiderivatives.

(a)
${\displaystyle \int x^2 \sin x ~dx}$

(b)
${\displaystyle \int {x - 1 \over x^2 + 1}~dx}$

(c)
${\displaystyle \int e^{-x} \sin (2x) ~dx}$

3.
(30 points) The function f is given by the table of values below.


\begin{displaymath}\begin{array}{c\vert c\vert c\vert c\vert c\vert c\vert c\ver... ...(x) & 1 & 0.96 & 0.84 & 0.66 & 0.45 & 0.24 & 0.05 \end{array}\end{displaymath}

Approximate ${\displaystyle \int _0^3 f(x) ~dx}$ by using

(a)
the left sum with 3 subintervals

(b)
the trapezoid rule with 3 subintervals


(c)
Simpson's rule with 6 subintervals.


4.
(10 points) Define the function F by ${\displaystyle F(x)=\int _0^x {\sin t \over t}~dt}$.(Since the integrand ${\displaystyle {\sin t \over t}}$ is undefined when t=0, we define its value to be 1 when t=0.)

(a)
Find the y-intercept of F.

(b)
Find all critical points of F in the interval $[ 0, 3\pi]$ (i.e., all values of x with F'(x)=0, $0 \leq x \leq 3\pi$).



Tony Phillips
2000-05-02