To Infinity

Infinitely finite areas

Sometimes a function will approach infinity, but the area under its curve will still be finite. Consider ​
${\int }_1^{\mathrm{\infty }}\frac{1}{x^2}dx$

There are actually two ways to solve this integral. The first method is just brute force. We evaluate the integral as though ∞ was a really large number. This method is useful if we need to do an integral in our heads.

The 2nd method is more elegant and mathematically correct. It involves setting up the integral as a limit by replacing ∞ with another variable such as t and finding the integral as $t\to \mathrm{\infty }$.

Example Problem 8.2.1

Evaluate ${\int }_{\frac{1}{3}}^{\mathrm{\infty }}\frac{1}{x^2}dx$

Cool, huh? But perhaps you might have to see it to believe it! (See Interactive 8.2.1)

Example Problem 8.2.2

Evaluate ${\int }_1^{\mathrm{\infty }}\frac{x^2+3x+6}{x^4}dx$

For a >0, the improper integral ∫_a^∞ x^q dx converges exactly when q <−1. The exponent need not be an integer: x^−3/2 also has a finite integral. At q =−1, the antiderivative ln x grows without bound; at q >−1, the power antiderivative grows without bound.