MATH 106B,C - CALCULUS II FALL 2011
QUIZ 7
NAME:
Show ALL your work CAREFULLY.
(a) Evaluate the improper integral
Z0
−∞
xe−x2dx.
The indefinite integral Rxe−x2dx can be evaluated using a simple substitution u=x2.
Thus, Zxe−x2dx =1
2Ze−udu
=−1
2e−x2+C.
Thus,
Z0
−∞
xe−x2dx = lim
b→−∞
−1
2e−x2
0
b
= lim
b→−∞ −1
2−e−b2=−1
2.
(b) Determine if the following improper integral converges or diverges. Justify your answer.
Z1
0
ex
x2dx
[Hint: use comparison test.]
For 0≤x≤1,ex≥1. It follows that
Z1
0
ex
x2dx ≥Z1
0
1
x2dx.
Since R1
0
1
x2dx diverges using the p-test (here p= 2 >1), it follows that the improper
integral R1
0
ex
x2dx must also diverge.
Date: November 2, 2011.
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