Barrons AP Calculus

Example 30 __

Determine whether or not converges.

SOLUTION: Although there is no elementary function whose derivative is e−x^2 ,
we can still show that the given improper integral converges. Note, first, that if x

1    then    x^2 x,  so  that    −x^2 −x     and     e−x

2

e−x.     Furthermore,   

.

Since converges and converges by the Comparison

Test.

Example 31 __

Show that converges.

SOLUTION: ;

we will use the Comparison Test to show that both of these integrals converge.
Since if , it follows that

.

We know that converges; hence must converge. Further, if x 1

then x + x^4 x^4 and , so .

We know that converges, hence also converges.

Thus the given integral, , converges.

NOTE: Examples 32 and 33 involve finding the volumes of solids. Both lead to
improper integrals.

BC  ONLY