Problem 30 - Essential Calculus

Question

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. $$ \int_{0}^{6} \frac{4}{\sqrt{6-x}} d x $$

Step-by-Step Solution

Answer

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

Step 1: 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.

Step 2: 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 \(F(x) = -8\sqrt{u} = -8\sqrt{6-x}\).

Step 3: 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.

Step 4: 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.