What is the volume of the solid of revolution of a parabolic segment bounded by the parabola ﻿y squared equals x minus 5﻿ and the line ﻿x equals 8﻿?

To find the volume of the solid of revolution generated by revolving a parabolic segment around the \(x\)-axis, we use the method of disks. The solid is formed by revolving the area bounded by the parabola and the line around the \(x\)-axis.

### Step 1: Understanding the equations
You are given the parabola:
\[
y^2 = x - 5
\]
This can be rewritten as:
\[
x = y^2 + 5
\]
The line given is \(x = 8\).

### Step 2: Determine the limits of integration
We need to determine the limits for \(y\). The boundary occurs when the parabola intersects the line \(x = 8\). To find the corresponding \(y\)-values, substitute \(x = 8\) into the equation of the parabola:
\[
y^2 = 8 - 5 = 3
\]
Thus, \(y = \pm \sqrt{3}\).

The limits for \(y\) are from \(-\sqrt{3}\) to \(\sqrt{3}\).

### Step 3: Set up the volume integral
Using the disk method, the formula for the volume is:
\[
V = \pi \int_{y_1}^{y_2} [R(y)]^2 \, dy
\]
where \(R(y)\) is the radius of each disk, which in this case is the \(x\)-coordinate of the parabola.

From the equation \(x = y^2 + 5\), the radius \(R(y) = y^2 + 5\).

### Step 4: Calculate the volume
The volume integral becomes:
\[
V = \pi \int_{-\sqrt{3}}^{\sqrt{3}} (y^2 + 5)^2 \, dy
\]

### Step 5: Solve the integral
First, expand \((y^2 + 5)^2\):
\[
(y^2 + 5)^2 = y^4 + 10y^2 + 25
\]

Now, integrate each term separately:
\[
V = \pi \int_{-\sqrt{3}}^{\sqrt{3}} (y^4 + 10y^2 + 25) \, dy
\]

Since the function is symmetric about the \(y\)-axis, we can double the integral from 0 to \(\sqrt{3}\):
\[
V = 2\pi \int_0^{\sqrt{3}} (y^4 + 10y^2 + 25) \, dy
\]

Now, compute the integral term by term:
\[
\int_0^{\sqrt{3}} y^4 \, dy = \left[ \frac{y^5}{5} \right]_0^{\sqrt{3}} = \frac{(\sqrt{3})^5}{5} = \frac{9\sqrt{3}}{5}
\]
\[
\int_0^{\sqrt{3}} 10y^2 \, dy = \left[ \frac{10y^3}{3} \right]_0^{\sqrt{3}} = \frac{10(\sqrt{3})^3}{3} = 10\sqrt{3}
\]
\[
\int_0^{\sqrt{3}} 25 \, dy = 25y \Big|_0^{\sqrt{3}} = 25\sqrt{3}
\]

### Step 6: Combine the results
Now, summing up the integrals:
\[
V = 2\pi \left( \frac{9\sqrt{3}}{5} + 10\sqrt{3} + 25\sqrt{3} \right)
\]
Factor out \(\sqrt{3}\):
\[
V = 2\pi \cdot \sqrt{3} \left( \frac{9}{5} + 10 + 25 \right)
\]
Simplify inside the parentheses:
\[
V = 2\pi \cdot \sqrt{3} \left( \frac{9 + 50 \cdot 5}{5} \right)
\ \ So \(V simplifiable reasonable.result sum_russian0 output

Explore how to calculate the volume of the solid of revolution formed by revolving a parabolic segment around the x-axis. This step-by-step guide uses the disk method to solve the problem of the region bounded by the parabola y² = x - 5 and the line x = 8. Suitable for students in Grades 11-12 studying calculus.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation