Problem 36 - 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. $$ \int_{0}^{2} \frac{1}{4-x^{2}} d x $$

Step-by-Step Solution

Answer

The improper integral converges and the evaluated result is \( \frac{\pi}{4} \).

Step 1: Identify the singularity

Identify that the singularity of the function \( f(x) = \frac{1}{4-x^{2}} \) is at x = 2. This makes the integral improper.

Step 2: Rewrite the integral

Rewrite the integral in limit form around the singularity: \( \lim_{b\to2^-} \int_{0}^{b} \frac{1}{4-x^{2}} dx \)

Step 3: Compute the indefinite integral

Compute the indefinite integral of the function. In this case, the antiderivative is \( \frac{1}{2} \arctan(\frac{x}{2}) \).

Step 4: Compute the limit as b approaches 2

Evaluate the integral by computing the limit as b approaches 2 from the left. This gives \( \lim_{b\to2^-} [\frac{1}{2} \arctan(\frac{b}{2}) - \frac{1}{2} \arctan(0)] = \frac{\pi}{4} \).

Step 5: Conclusion

Since the limit exists and is finite, the integral converges and its value is \( \frac{\pi}{4} \). It can be confirmed by the graphing utility.