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

Jasonbradley

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Jun 16, 2024

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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.