2\left(12-y\right)\sqrt{81-y^2} integrateb from -9 to 9

To solve the integral of \( 2(12 - y) \sqrt{81 - y^2} \) with respect to \(y\), from \(-9\) to \(9\), let's follow these steps:

### Step 1: Expression Setup
We are tasked with computing:
\[
\int_{-9}^{9} 2(12 - y) \sqrt{81 - y^2} \, dy
\]
Here, the square root term \( \sqrt{81 - y^2} \) suggests that a trigonometric substitution may simplify the expression. Notice that \(81 - y^2\) resembles \(81(1 - \frac{y^2}{81})\), which is a form suitable for the trigonometric substitution \(y = 9\sin(\theta)\).

### Step 2: Trigonometric Substitution
Let’s apply the substitution:
\[
y = 9\sin(\theta)
\]
This gives:
\[
dy = 9\cos(\theta) \, d\theta
\]
Substituting into the square root expression:
\[
\sqrt{81 - y^2} = \sqrt{81 - 81\sin^2(\theta)} = 9\cos(\theta)
\]

### Step 3: Limits of Integration
When \( y = -9 \), we have \( \theta = -\frac{\pi}{2} \), and when \( y = 9 \), we have \( \theta = \frac{\pi}{2} \).

### Step 4: Substituting into the Integral
Now substitute everything into the integral:
\[
\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} 2(12 - 9\sin(\theta)) \cdot 9\cos(\theta) \cdot 9\cos(\theta) \, d\theta
\]
Simplifying the expression:
\[
2 \cdot 81 \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} (12 - 9\sin(\theta)) \cos^2(\theta) \, d\theta
\]
\[
= 162 \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} (12 - 9\sin(\theta)) \cos^2(\theta) \, d\theta
\]

### Step 5: Break Into Two Integrals
Now, we can break the integral into two parts:
\[
162 \left[ 12 \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \cos^2(\theta) \, d\theta - 9 \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sin(\theta) \cos^2(\theta) \, d\theta \right]
\]

#### First Integral: \( \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \cos^2(\theta) \, d\theta \)
Using the identity \( \cos^2(\theta) = \frac{1 + \cos(2\theta)}{2} \), we get:
\[
\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \cos^2(\theta) \, d\theta = \frac{1}{2} \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} (1 + \cos(2\theta)) \, d\theta = \frac{\pi}{2}
\]

#### Second Integral: \( \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sin(\theta) \cos^2(\theta) \, d\theta \)
For this, we use a substitution. Let \( u = \cos(\theta) \), so \( du = -\sin(\theta) d\theta \). The limits change accordingly.

### Step 6: Compute and Combine Results
The first integral gives \( 162 \cdot 12 \cdot \frac{\pi}{2} \), and the second integral will evaluate to zero because it is an odd function integrated symmetrically about zero.

Thus, the total result of the integral is:
\[
162 \cdot 12 \cdot \frac{\pi}{2} = 972\pi
\]

### Final Answer:
\[
\int_{-9}^{9} 2(12 - y) \sqrt{81 - y^2} \, dy = 972\pi
\]

Would you like further details or clarifications?

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**Related Questions:**
1. What is the general approach to trigonometric substitution in integrals?
2. How can we determine when a function's integral over a symmetric interval will be zero?
3. What other substitution methods could simplify square root terms?
4. Why does the integral of an odd function over a symmetric interval equal zero?
5. How does the Pythagorean identity help in solving integrals involving square roots?

**Tip:** Recognizing the form of a function (like \(a^2 - x^2\)) can guide you toward an appropriate trigonometric substitution to simplify the integration process.

2(12 - y)\sqrt{81 - y^2} integrate from -9 to 9

Explore how to solve the definite integral of 2(12 - y)√(81 - y²) from -9 to 9 using trigonometric substitution. This detailed guide walks through key calculus concepts, including substitution methods and integral evaluation, suitable for Grades 11-12 and college calculus students.

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