List of Notes - Category: Improper Integrals
When we defined definite integral `int_a^bf(x)dx` we assumed that interval [a,b] is is finite and that function f doesn't have infinite discontinuities. Let's extend integral to the case where interval [a,b] is infinite.
Suppose that f is a positive continuous function defined on a finite interval [a,b) but has a vertical asymptote at b. Let S be the unbounded region under the graph of f and above the x-axis between a and b. (For Type 1 integrals, the regions extended indefinitely in a horizontal direction. Here the region is infinite in a vertical direction.) The area of the part of S between a and t is `A(t)=int_a^t f(x)dx` .
Sometimes it is impossible to find the exact value of an improper integral and yet it is important to know whether it is convergent or divergent. In such cases the following test is useful. Although we state it for Type 1 integrals, a similar theorem is true for Type 2 integrals.