=√√9 = Find the volume of the solid with the semicircle base y sections perpendicular to the x-axis are squares. 2 x and the cross
=√√9 = Find the volume of the solid with the semicircle base y sections perpendicular to the x-axis are squares. 2 x and the cross
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![**Problem Statement:**
Find the volume of the solid with the semicircle base \( y = \sqrt{9 - x^2} \) and the cross sections perpendicular to the \( x \)-axis are squares.
**Explanation:**
1. **Base Shape:**
- The base of the solid is a semicircle described by the equation \( y = \sqrt{9 - x^2} \). This semicircle is centered at the origin on the coordinate plane and has a radius of 3.
2. **Cross Sections:**
- The cross sections perpendicular to the \( x \)-axis are squares. This means that for every \( x \) value within the semicircle’s domain, there is a square with sides equal to the \( y \)-value at that point.
3. **Volume Calculation:**
- To find the volume of the solid, one would integrate the area of the squares along the \( x \)-axis over the range \([-3, 3]\). The side length of each square is \( 2y = 2\sqrt{9 - x^2} \), and consequently, the area is \((2\sqrt{9 - x^2})^2 = 4(9 - x^2)\).
The entire volume \( V \) is then calculated by integrating the area from \( x = -3 \) to \( x = 3 \):
\[ V = \int_{-3}^{3} 4(9 - x^2) \, dx \]
This integral represents the sum of the volumes of these infinitesimally thin square slices that make up the solid.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5eb09660-693f-4b0a-9c98-8e645b8ba753%2Ff830c60e-0da2-4f2d-a0b7-d97ff43b351f%2Flogyk3_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the volume of the solid with the semicircle base \( y = \sqrt{9 - x^2} \) and the cross sections perpendicular to the \( x \)-axis are squares.
**Explanation:**
1. **Base Shape:**
- The base of the solid is a semicircle described by the equation \( y = \sqrt{9 - x^2} \). This semicircle is centered at the origin on the coordinate plane and has a radius of 3.
2. **Cross Sections:**
- The cross sections perpendicular to the \( x \)-axis are squares. This means that for every \( x \) value within the semicircle’s domain, there is a square with sides equal to the \( y \)-value at that point.
3. **Volume Calculation:**
- To find the volume of the solid, one would integrate the area of the squares along the \( x \)-axis over the range \([-3, 3]\). The side length of each square is \( 2y = 2\sqrt{9 - x^2} \), and consequently, the area is \((2\sqrt{9 - x^2})^2 = 4(9 - x^2)\).
The entire volume \( V \) is then calculated by integrating the area from \( x = -3 \) to \( x = 3 \):
\[ V = \int_{-3}^{3} 4(9 - x^2) \, dx \]
This integral represents the sum of the volumes of these infinitesimally thin square slices that make up the solid.
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