Barrons AP Calculus - David Bock

Since converges and dx 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 0 < x 1,

then x + x^4 > x and 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

EXAMPLE 32

Find the volume, if it exists, of the solid generated by rotating the region in the first quadrant

bounded above by at the left by x = 1, and below by y = 0, about the x-axis.