3.6 Q-19

Question Statement

Evaluate the integral:

$$\int_{0}^{\frac{\pi}{3}} \cos^2 \theta \sin \theta , d\theta$$

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Background and Explanation

To solve this integral, we use a substitution method. The integral contains both $\cos^2 \theta$ and $\sin \theta$, which suggests we can make a substitution to simplify the expression. Specifically, we’ll use a substitution to eliminate the product of trigonometric functions and solve for the integral step by step.

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Solution

Start by recognizing that the integral involves a product of $\cos^2 \theta$ and $\sin \theta$. We can use substitution to simplify the integral.

1. Substitute: Let $u = \cos \theta$. Then, the derivative of $u$ with respect to $\theta$ is $du = -\sin \theta , d\theta$.

   This transforms the integral as follows:

$$\int_{0}^{\frac{\pi}{3}} \cos^2 \theta \sin \theta , d\theta = - \int_{0}^{\frac{\pi}{3}} \cos^2 \theta (-\sin \theta) , d\theta$$

This simplifies to:

$$= - \int_{0}^{\frac{\pi}{3}} u^2 , du$$

2. Change the limits: When $\theta = 0$, $u = \cos 0 = 1$, and when $\theta = \frac{\pi}{3}$, $u = \cos \frac{\pi}{3} = \frac{1}{2}$. So, the limits of integration change from $0$ to $\frac{\pi}{3}$ for $\theta$, to $1$ to $\frac{1}{2}$ for $u$.

3. Integrate: Now, we integrate $u^2$ with respect to $u$:

$$\int u^2 , du = \frac{u^3}{3}$$

Thus, the integral becomes:

$$- \left[ \frac{u^3}{3} \right]_{1}^{\frac{1}{2}}$$

4. Evaluate the limits: Substitute the limits into the result:

$$= - \left[ \frac{\left( \frac{1}{2} \right)^3 - (1)^3}{3} \right]$$

Simplify the expression:

$$= - \left[ \frac{\frac{1}{8} - 1}{3} \right] = - \frac{1}{3} \left[ \frac{1}{8} - 1 \right]$$ $$= - \frac{1}{3} \left[ \frac{1 - 8}{8} \right] = - \frac{1}{3} \times \frac{-7}{8}$$

5. Final Answer: Simplifying this, we get:

$$= \frac{7}{24}$$

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Key Formulas or Methods Used

• Substitution: Let $u = \cos \theta$, so $du = -\sin \theta , d\theta$.
• Integral of a power: $\int u^2 , du = \frac{u^3}{3}$.

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Summary of Steps

1. Use the substitution $u = \cos \theta$, and convert the integral.
2. Change the limits of integration according to $u = \cos \theta$.
3. Integrate $u^2$ with respect to $u$.
4. Substitute the limits into the integral and simplify.
5. The final answer is:

$$\frac{7}{24}$$