Evaluating the Definite Integral
In this article, we will evaluate the following definite integral:
$\frac{\int_{0}^{a} 4x^4 \sqrt{a^2 - x^2} dx}{\int_{0}^{a} ax^2 \sqrt{a^2 - x^2} dx}$
To evaluate this integral, we will first simplify the numerator and denominator separately.
Numerator
Let's evaluate the numerator:
$\int_{0}^{a} 4x^4 \sqrt{a^2 - x^2} dx$
We can start by substituting $x = a \sin \theta$, which implies that $dx = a \cos \theta d\theta$. Then, we have:
$\int_{0}^{a} 4x^4 \sqrt{a^2 - x^2} dx = \int_{0}^{\frac{\pi}{2}} 4a^4 \sin^4 \theta \sqrt{a^2 - a^2 \sin^2 \theta} a \cos \theta d\theta$
Simplifying, we get:
$\int_{0}^{\frac{\pi}{2}} 4a^5 \sin^4 \theta \cos^2 \theta d\theta$
Now, we can use the following trigonometric identities:
- $\sin^2 \theta = \frac{1 - \cos 2\theta}{2}$
- $\cos^2 \theta = \frac{1 + \cos 2\theta}{2}$
Substituting these identities, we get:
$\int_{0}^{\frac{\pi}{2}} 4a^5 \left( \frac{1 - \cos 2\theta}{2} \right)^2 \left( \frac{1 + \cos 2\theta}{2} \right) d\theta$
Simplifying and integrating, we get:
$\int_{0}^{\frac{\pi}{2}} 4a^5 \left( \frac{3 - 2\cos 2\theta - \cos^2 2\theta}{8} \right) d\theta$
Evaluating the integral, we get:
$\frac{a^5 \pi}{8}$
Denominator
Now, let's evaluate the denominator:
$\int_{0}^{a} ax^2 \sqrt{a^2 - x^2} dx$
Using the same substitution $x = a \sin \theta$, we get:
$\int_{0}^{a} ax^2 \sqrt{a^2 - x^2} dx = \int_{0}^{\frac{\pi}{2}} a^3 \sin^2 \theta \sqrt{a^2 - a^2 \sin^2 \theta} a \cos \theta d\theta$
Simplifying, we get:
$\int_{0}^{\frac{\pi}{2}} a^4 \sin^2 \theta \cos^2 \theta d\theta$
Using the same trigonometric identities, we get:
$\int_{0}^{\frac{\pi}{2}} a^4 \left( \frac{1 - \cos 2\theta}{2} \right) \left( \frac{1 + \cos 2\theta}{2} \right) d\theta$
Simplifying and integrating, we get:
$\int_{0}^{\frac{\pi}{2}} a^4 \left( \frac{1 - \cos^2 2\theta}{4} \right) d\theta$
Evaluating the integral, we get:
$\frac{a^4 \pi}{8}$
Final Answer
Now, we can evaluate the original expression:
$\frac{\int_{0}^{a} 4x^4 \sqrt{a^2 - x^2} dx}{\int_{0}^{a} ax^2 \sqrt{a^2 - x^2} dx} = \frac{\frac{a^5 \pi}{8}}{\frac{a^4 \pi}{8}} = \boxed{a}$
Therefore, the final answer is $a$.