## Evaluating the Definite Integral Ratio

This article will explore the evaluation of the following definite integral ratio:

**(int_(0)^(a)4x^(4)sqrt(a^(2)-x^(2))dx)/(int_(0)^(a)(ax)^(2)sqrt(a^(2)-x^(2))dx)=**

Let's break down the process step by step.

### Simplifying the Expression

First, we can simplify the expression by noting that:

- The denominator has a constant factor of 'a²'. We can pull this out:

**(int_(0)^(a)4x^(4)sqrt(a^(2)-x^(2))dx)/(a² * int_(0)^(a)x^(2)sqrt(a^(2)-x^(2))dx)=**

- Now, we can focus on evaluating the two integrals separately.

### Evaluating the Integrals

We'll use trigonometric substitution to evaluate both integrals.

**Integral 1: int_(0)^(a)4x^(4)sqrt(a^(2)-x^(2))dx**

**Substitution:**Let x = a sin(θ), then dx = a cos(θ) dθ.**Limits:**When x = 0, θ = 0. When x = a, θ = π/2.

Now the integral becomes:

int_(0)^(π/2) 4a^4 sin^4(θ) * a cos(θ) * a cos(θ) dθ

Simplifying:

4a^6 * int_(0)^(π/2) sin^4(θ) cos^2(θ) dθ

We can evaluate this integral using integration by parts or by using the reduction formula for integrals of powers of sine and cosine. The result will be a function of 'a'.

**Integral 2: int_(0)^(a)x^(2)sqrt(a^(2)-x^(2))dx**

**Substitution:**The same substitution as before, x = a sin(θ).**Limits:**Remain the same.

The integral becomes:

int_(0)^(π/2) a^2 sin^2(θ) * a cos(θ) * a cos(θ) dθ

Simplifying:

a^4 * int_(0)^(π/2) sin^2(θ) cos^2(θ) dθ

Again, this integral can be evaluated using integration techniques.

### Combining the Results

After evaluating both integrals, we will have expressions in terms of 'a'. Substitute these expressions back into the original ratio:

**(4a^6 * Integral 1 result) / (a² * a^4 * Integral 2 result)**

Simplifying further, we get:

**(4 * Integral 1 result) / (Integral 2 result)**

This will give us the final result, which will be a numerical value or an expression depending on the values of the integrals.

### Important Notes:

- Remember to carefully consider the limits of integration when performing the substitution.
- You will need to apply integration techniques to evaluate both integrals.
- The final result may involve trigonometric functions of 'a' or constants.

By following these steps, you can effectively evaluate the given definite integral ratio.