2*π*∫−2−1​(2*y*+4)21+(8*y*+16)2​*dy*

The given integral is:

\[
2\pi \int_{-2}^{-1} \left( 2y + 4 \right) \cdot \frac{2}{1 + \left(8y + 16\right)^2} \, dy
\]

We will now solve this step by step:

### Step 1: Simplify the expression

We can start by simplifying the expression inside the integral. First, notice that the term \( 8y + 16 \) can be rewritten as:

\[
8y + 16 = 8(y + 2)
\]

Thus, the integral becomes:

\[
2\pi \int_{-2}^{-1} \left( 2y + 4 \right) \cdot \frac{2}{1 + [8

Learn how to solve the definite integral 2π ∫ from -2 to -1 of (2y + 4) / (1 + (8y + 16)^2) dy using trigonometric substitution and simplification. This problem covers key concepts in integral calculus and trigonometric substitution. Suitable for undergraduate calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation