Problem 26 - College Algebra and Calculus: An Applied Approach

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. $$ \int_{0}^{2} \frac{x}{\sqrt{4-x^{2}}} d x $$

Step-by-Step Solution

Answer

The improper integral \(\int_{0}^{2} \frac{x}{\sqrt{4-x^{2}}} dx \) converges and the value can be obtained by closely following the described steps. Do not forget to verify the answer with a symbol computation system or graphing utility.

Step 1: Re-Write the Integral

Firstly, rewrite the integral \( \int_{0}^{2} \frac{x}{\sqrt{4-x^{2}}} dx \) as a limit problem by substituting \( 2 \) with \( b \) and taking the limit as \( b \) approaches \( 2 \) from the left side: \( \lim_{b \to 2^{-}} \int_{0}^{b} \frac{x}{\sqrt{4-x^{2}}} dx \).

Step 2: Apply Substitution

For evaluation, apply substitution method by setting \( u = 4 - x^{2} \). Hence, find \( du = -2x dx \) and replace \( x dx \) with \( -1/2 du \). The limits also change according to the substitution, with the lower limit \( u(0) = 4 \) and the upper limit \( u(b) = 4 - b^{2} \). Now, the integral becomes: \( -1/2 \lim_{b \to 2^{-}} \int_{4}^{4-b^2} \frac{1}{\sqrt{u}} du \). Notice that the limits are flipped so the negative sign will be cancelled.

Step 3: Integrate

The integral \( \int \frac{1}{\sqrt{u}} du \) can be integrated to \( 2\sqrt{u} \). Substituting the limits back gives \( \lim_{b \to 2^{-}} [2\sqrt{4 - b^{2}} - 2\sqrt{4}] \).

Step 4: Calculate the Limit

Finally, evaluate the limit as \( b \) approaches \( 2 \) from left to get the result.

Step 5: Check the Results

Lastly, verify the results with a graphing utility or a symbol computation system, and confirm if the integration was correct according to them.