10 3 Integrals with Infinite Limits of Integration

10. 3 Integrals with Infinite Limits of Integration Rita Korsunsky

Integrals with Infinite Limits of Integration are Improper Integrals Definition: i) If f is continuous on [a, ), then ii) If f is continuous on (- , a], then provided the limit exists. An improper integral is said to converge if the limit exists, and if it does not exist, the integral is said to diverge.

Example 1 Determine whether the integral converges or diverges, and if it converges, find its value. (converges) (diverges)

If area under a curve is infinite (integral diverges), then volume of solid of revolution when you rotate this region around x or y-axis does not have to be infinite. (Integral could converge) Diverges from example 1 part b Converges from example 1 part a

If area under a curve is a constant (integral converges), then volume of solid of revolution when you rotate this finite region around x or y-axis does not have to be constant. (Integral could diverge) Converges from example 1 part a Diverges from Converges example 1 part b

Example 2 Assign an area to the region that lies under the graph of y=ex, over the x-axis, and to the left of x=1.

Another Form of Improper Integral Let f be continuous for every x. If a is any real number, then provided both of the improper integrals on the right converge.

Example 3 Evaluate the following: First evaluate this: Similarly: Add both values: