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Chapter 6: Problem 30

Determine whether the improper integral diverges or converges. Evaluate the integral if it converges, and check your results with the results obtained by using the integration capabilities of a graphing utility. $$ \int_{0}^{6} \frac{4}{\sqrt{6-x}} d x $$

Short Answer

Expert verified
The improper integral, \(\int_{0}^{6} \frac{4}{\sqrt{6-x}} dx\), diverges.

Step by step solution

01

Rewrite the Integral

This integral is rewritten as \(\lim _{t \rightarrow 6^{-}} \int_{0}^{t} \frac{4}{\sqrt{6-x}} dx\). Now it's clear that as \(x\) approaches \(6\) from the left, the integral becomes infinite because division by zero is undefined.
02

Find the Antiderivative

The antiderivative of \(\frac{4}{\sqrt{6-x}}\) can be determined by applying substitution, choosing \(u = 6 - x\), which gives \(du = -dx\). The antiderivative then becomes \(-4 \int \frac{1}{\sqrt{u}} du\) and further integration gives \(F(x) = -8\sqrt{u} = -8\sqrt{6-x}\).
03

Apply the Fundamental Theorem of Calculus

The upper and lower limits of the integral are substituted into the antiderivative function. This gives \(-8\sqrt{6 - t}\) for the upper limit and \(-8\sqrt{6 - 0}\) for the lower limit. By taking limit as \(t\) approaches \(6^{-}\), the expression becomes \(8\sqrt{6} - 8\sqrt{6 - t}\). The integral can now be evaluated.
04

Evaluate the Integral

Evaluating this limit, the integral \(\lim _{t \rightarrow 6^{-}} 8\sqrt{6} - 8\sqrt{6 - t}\) approaches infinity, indicating that the convergence test fails. Therefore the improper integral diverges.

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Most popular questions from this chapter

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