Q.1.a) - Calculus

Question

If $0\le f\left(x\right) \le g\left(x\right)$for every x$\ge$0 and the improper integral${\int }_0^{\infty }g\left(x\right)dx$ converges, then the improper integral ${\int }_0^{\infty }f\left(x\right)dx$ converges. The objective is to  whether determine the statement is true or false

Step-by-Step Solution

Answer

True

Step 1: Understand the Problem

We are given that \(0 \leq f(x) \leq g(x)\) for every \(x \geq 0\) and need to analyze the relationship between the improper integrals of \(f\) and \(g\).

Step 2: Apply the Comparison Test for Improper Integrals

The Comparison Test states that if \(0 \leq f(x) \leq g(x)\) for all \(x \geq a\), then:

• If \(\int_a^{\infty} g(x)\,dx\) converges, then \(\int_a^{\infty} f(x)\,dx\) also converges.
• If \(\int_a^{\infty} f(x)\,dx\) diverges, then \(\int_a^{\infty} g(x)\,dx\) also diverges.

Step 3: State the Result

True