Question

Evaluate \(\int_{0}^{a}\left(a^{2}-x^{2}\right)^{5 / 2} d x\)

Short answer

Question: Evaluate the integral \(\int_{0}^{a}(a^2-x^2)^{5/2}dx\). Answer: \(\frac{8}{15}a^6\)

Step-by-step solution

Identify the integrand and limits of integration

In the given integral, we have the integrand as \((a^2 - x^2)^{5/2}\) and the limits of integration as \(x=0\) to \(x=a\).

Identify techniques or methods for simplifying the integral

Since the integrand is not easily integrable as given, we can try substitution. Let \(x = a\sin(\theta)\), with \(0\leq\theta\leq\frac{\pi}{2}\). Then, \(dx = a\cos(\theta)d\theta\) and \(a^2 - x^2 = a^2(1-\sin^2(\theta)) = a^2\cos^2(\theta)\).

Perform the integration

We can rewrite the integral as: \(\int_{0}^{a}(a^{2}-x^{2})^{5 / 2} d x = \int_{0}^{\frac{\pi}{2}}(a^2\cos^2(\theta))^{\frac{5}{2}}\cdot a\cos(\theta) d\theta = a^6\int_{0}^{\frac{\pi}{2}}\cos^6(\theta)\cdot \cos(\theta)d\theta = a^6\int_{0}^{\frac{\pi}{2}}\cos^7(\theta)d\theta\) Now, to solve this integral, let's use the reduction formula: \(\int\cos^n(\theta)d\theta = \frac{1}{n}\cos^{n-1}(\theta)\sin(\theta) + \frac{n-1}{n}\int\cos^{n-2}(\theta)d\theta\) Applying the reduction formula to \(\cos^7(\theta)\): \(\int\cos^7(\theta)d\theta = \frac{1}{7}\cos^6(\theta)\sin(\theta) + \frac{6}{7}\int\cos^5(\theta)d\theta\) Applying the reduction formula once again to \(\cos^5(\theta)\): \(\int\cos^5(\theta)d\theta = \frac{1}{5}\cos^4(\theta)\sin(\theta) + \frac{4}{5}\int\cos^3(\theta)d\theta\) And finally, applying it to \(\cos^3(\theta)\): \(\int\cos^3(\theta)d\theta = \frac{1}{3}\cos^2(\theta)\sin(\theta) + \frac{2}{3}\int\cos(\theta)d\theta\) Combining these results, we have: \(\int\cos^7(\theta)d\theta = \frac{1}{7}\cos^6(\theta)\sin(\theta) + \frac{6}{7}\left(\frac{1}{5}\cos^4(\theta)\sin(\theta) + \frac{4}{5}\left(\frac{1}{3}\cos^2(\theta)\sin(\theta) + \frac{2}{3}\int\cos(\theta)d\theta\right)\right)\) Now, we can integrate the remaining part: \(\int\cos(\theta)d\theta = \sin(\theta)\)

Evaluate the integral over the given limits

Plugging the integral back into our original expression and evaluating the limits, we get: \(\int_{0}^{\frac{\pi}{2}}\cos^7(\theta)d\theta = a^6\left[\frac{1}{7}\cos^6(\theta)\sin(\theta) + \frac{6}{7}\left(\frac{1}{5}\cos^4(\theta)\sin(\theta) + \frac{4}{5}\left(\frac{1}{3}\cos^2(\theta)\sin(\theta) + \frac{2}{3}\sin(\theta)\right)\right)\right]_{0}^{\frac{\pi}{2}}\) Evaluating at the limits, we have: \(= a^6\left[\frac{1}{7}\cdot 0\cdot 1 + \frac{6}{7}\left(\frac{1}{5}\cdot 0\cdot 1 + \frac{4}{5}\left(\frac{1}{3}\cdot 1\cdot 1 + \frac{2}{3}\cdot 1\right)\right) - 0\right]\) After simplification, we get the final answer: \(\int_{0}^{a}(a^{2}-x^{2})^{5 / 2} d x = \frac{8}{15}a^6\)