Calculating Volume by Rotating a Region

Jul 10, 2024

Calculating Volume by Rotating a Region

Methods Overview

Disc Method
Washer Method

Disc Method Explained

Concept: Calculate volume by rotating a curve around an axis to form a disc.
Steps:
1. Take cross-sectional area (circle).
2. Rotate around axis to form a disc shape (similar to a cylinder).
3. Volume of cylinder = Cross-sectional area × Height = πr²h
4. To find volume of a solid: Sum of volumes of infinitesimal discs from a to b.
Volume Formula:

$\pi \int_a^b [r(x)]^2 dx$

Rotating About the X-Axis

Define limits a and b (in terms of x).
Volume formula:

$\pi \int_a^b [r(x)]^2 dx$

Rotating About the Y-Axis

Define limits c and d (in terms of y).
Volume formula:

$\pi \int_c^d [r(y)]^2 dy$

Example: Rotating y = √x about the X-Axis

Given: Function y = √x, bounded between x = 0 and x = 4.
Steps:
1. Graph the function.
2. Determine radius r(x) = √x.
3. Set up integral:
$\pi \int_0^4 [r(x)]^2 dx = \pi \int_0^4 x dx$ 4. Integrate:

$[\frac{x^2}{2}] 0 ^4$

Example: Rotating y = 1/x about the X-Axis

Given: Function y = 1/x, bounded between x = 1 and x = 3.
Steps:
1. Graph the function.
2. Determine radius r(x) = 1/x.
3. Set up integral:
$\pi \int_1^3 [\frac{1}{x}]^2 dx = \pi \int_1^3 x^{-2} dx$ 4. Integrate:

$\pi [-1/3 - -[-1/1] = \frac{2}{3} \pi]$

Example: Rotating y = x² about the Y-Axis

Given: Function y = x², bounded between x = 0 and y = 4.
Steps:
1. Graph the function.
2. Determine radius, convert x in terms of y: x = √y.
3. Set up integral:
$\pi \int_0^4 [y] dy$ 4. Integrate:

$[\frac{y^2}{2}] 0 ^4$

Example: Rotating y = x^(2/3) about the Y-Axis

Given: Function y = x^(2/3), bounded between x = 0 and y = 1.
Steps:
1. Graph the function.
2. Determine radius, convert x in terms of y: x = y^(3/2).
3. Set up integral:
$\pi \int_0^1 [y^{3}] dy$ 4. Integrate:

$[\frac{y^4}{4}] 0 ^1 = \frac{\pi}{4}$

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