If f(x) is a continuous function such that int_(a)^(oo)f(x)dx diverges and a < b, then int_(b)^(oo)f(x)dx also diverges. True False Next Question

Properties of Integrals: Let's consider the properties of integrals and continuous functions. If the integral of a continuous function f(x) from a to infinity diverges, it means that the area under the curve from a to infinity is infinite. Since b is greater than a (a < b), the integral from b to infinity would be a part of the integral from a to infinity.Analysis of Divergence: Now, let's analyze the implication of the divergence of the integral from a to infinity. If the integral from a to infinity diverges, then adding or removing a finite part of the integral (from a to b) should not affect the divergence. This is because subtracting a finite value from infinity still leaves us with infinity.Implication of Divergence: Therefore, if the integral of f(x) from a to infinity diverges, the integral from b to infinity, which is a subset of the original integral, must also diverge. This is because the function is continuous, and the divergence from a to infinity indicates that the function does not settle into a pattern that would allow the integral to converge from any point onward.

Resources

Testimonials

Plans

Resources

Testimonials

Plans

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

If $f(x)$ is a continuous function such that $\int_{a}^{\infty}f(x)\,dx$ diverges and a < b , then $\int_{b}^{\infty}f(x)\,dx$ also diverges. $\newline$ True $\newline$ False $\newline$

View step-by-step help

Home

Math Problems

Algebra 2

Identify discrete and continuous random variables

Full solution

Q. If

f(x)

is a continuous function such that

\int_{a}^{\infty}f(x)\,dx

diverges and

a < b

, then

\int_{b}^{\infty}f(x)\,dx

also diverges.

\newline

True

\newline

False

\newline

Properties of Integrals: Let's consider the properties of integrals and continuous functions. $\newline$ If the integral of a continuous function $f(x)$ from $a$ to infinity diverges, it means that the area under the curve from $a$ to infinity is infinite. $\newline$ Since $b$ is greater than $a$ (a < b), the integral from $b$ to infinity would be a part of the integral from $a$ to infinity.
Analysis of Divergence: Now, let's analyze the implication of the divergence of the integral from $a$ to infinity. If the integral from $a$ to infinity diverges, then adding or removing a finite part of the integral (from $a$ to $b$ ) should not affect the divergence. This is because subtracting a finite value from infinity still leaves us with infinity.
Implication of Divergence: Therefore, if the integral of $f(x)$ from $a$ to infinity diverges, the integral from $b$ to infinity, which is a subset of the original integral, must also diverge. $\newline$ This is because the function is continuous, and the divergence from $a$ to infinity indicates that the function does not settle into a pattern that would allow the integral to converge from any point onward.