Problem 4 - College Algebra and Calculus: An Applied Approach

Question

Decide whether the integral is improper. Explain your reasoning. $$ \int_{1}^{\infty} x^{2} d x $$

Step-by-Step Solution

Answer

The integral \(\int_{1}^{\infty} x^{2} dx\) is an improper integral since the interval of integration is infinite.

Step 1: Identify the limits of integration and the function to be integrated

The given integral is \(\int_{1}^{\infty} x^{2} dx\). Here, the limits of integration are 1 and infinity (\infty), and the function to be integrated \(f(x) = x^{2}\).

Step 2: Determine if the integral is improper

An integral is improper if either the interval of integration is infinite, or the function to be integrated approaches infinity at one or more points in the interval of integration. In this case, the interval of integration is from 1 to infinity, which is an infinite interval. Therefore, this is an improper integral.