Let's solve the integral $\int_0^2 \sqrt{x(4-x)} \, dx$ using the substitution $x = 4\sin^2\theta$.

Step 1: Substitution

Given $x = 4\sin^2\theta$, we need to calculate the corresponding expressions for $dx$, the limits of integration, and simplify the integrand:

1. Substitute $x = 4\sin^2\theta$:

   • Then, $dx = 8\sin\theta\cos\theta \, d\theta = 4\sin(2\theta) \, d\theta$ (using the double-angle identity $\sin(2\theta) = 2\sin\theta\cos\theta$).

2. Change the limits of integration:

   • When $x = 0$, $4\sin^2\theta = 0 \implies \theta = 0$.
   • When $x = 2$, $4\sin^2\theta = 2 \implies \sin^2\theta = \frac{1}{2} \implies \theta = \frac{\pi}{4}$.

Step 2: Simplify the integrand

Now, we substitute into the integrand: $\sqrt{x(4-x)} = \sqrt{4\sin^2\theta(4 - 4\sin^2\theta)} = \sqrt{16\sin^2\theta\cos^2\theta} = 4\sin\theta\cos\theta = 2\sin(2\theta)$ Thus, the integral becomes: $\int_0^2 \sqrt{x(4-x)} \, dx = \int_0^{\frac{\pi}{4}} 2\sin(2\theta) \cdot 4\sin(2\theta) \, d\theta = 8 \int_0^{\frac{\pi}{4}} \sin^2(2\theta) \, d\theta$

Step 3: Solve the integral

We can use the identity $\sin^2(2\theta) = \frac{1 - \cos(4\theta)}{2}$ to simplify: $\int_0^{\frac{\pi}{4}} \sin^2(2\theta) \, d\theta = \frac{1}{2} \int_0^{\frac{\pi}{4}} (1 - \cos(4\theta)) \, d\theta$ This gives: [ \frac{1}{2} \left[ \theta - \frac{\sin(4\theta)}{4} \right]_0^{\frac{\pi}{4}} = \frac{1}{2} \left[ \frac{\pi}{4} - 0 \right] = \frac{\pi}{8} ]

Step 4: Final answer

Multiplying everything together, we get: $8 \cdot \frac{\pi}{8} = \pi$

Thus, the exact value of the integral is $\boxed{\pi}$.

Would you like any more details, or have any questions about this solution?

Here are 5 related questions:

1. How do trigonometric identities simplify integrals involving squares of sine or cosine functions?
2. What other types of substitutions are useful for integrals of this kind?
3. Can this method be used for other limits or different forms of the integrand?
4. How does the double-angle identity help simplify integrals?
5. How would the integral change if the limits of integration were different?

Tip: Always remember to change the limits of integration when switching variables!