Use an area formula from geometry to find the value of the integral by interpreting it as the (signed) area under the graph of an appropriately chosen function. S√64-x² dx 8 S√64-x² ax dx = (Type an exact answer, using as needed.)
Use an area formula from geometry to find the value of the integral by interpreting it as the (signed) area under the graph of an appropriately chosen function. S√64-x² dx 8 S√64-x² ax dx = (Type an exact answer, using as needed.)
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![### Topic: Geometric Interpretation of Integrals
**Instructions:**
Use an area formula from geometry to find the value of the integral by interpreting it as the (signed) area under the graph of an appropriately chosen function.
**Integral to Evaluate:**
\[ \int_{-8}^{8} \sqrt{64 - x^2} \, dx \]
**Note:**
(Type an exact answer, using \(\pi\) as needed.)
### Explanation:
The integral \[ \int_{-8}^{8} \sqrt{64 - x^2} \, dx \] represents the area under the function \( y = \sqrt{64 - x^2} \) from \( x = -8 \) to \( x = 8 \).
This function describes the upper semicircle of a circle with radius 8, centered at the origin \((0, 0)\). The area under this curve represents half of the area of a full circle with radius 8.
### Calculation:
The area \(A\) of a full circle with radius \(r\) is given by the formula:
\[ A = \pi r^2 \]
For a circle with radius 8:
\[ A = \pi (8)^2 = 64\pi \]
Since the integral only considers the upper half of the circle (a semicircle), the area under the function is half of the total area:
\[ \text{Area under the curve} = \frac{1}{2} \times 64\pi = 32\pi \]
Therefore,
\[ \int_{-8}^{8} \sqrt{64 - x^2} \, dx = 32\pi \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3d691669-9eb8-4a75-be21-eced36655373%2F67757a8d-04b5-4b6d-a398-0c5529020875%2F8jkxu2o_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Topic: Geometric Interpretation of Integrals
**Instructions:**
Use an area formula from geometry to find the value of the integral by interpreting it as the (signed) area under the graph of an appropriately chosen function.
**Integral to Evaluate:**
\[ \int_{-8}^{8} \sqrt{64 - x^2} \, dx \]
**Note:**
(Type an exact answer, using \(\pi\) as needed.)
### Explanation:
The integral \[ \int_{-8}^{8} \sqrt{64 - x^2} \, dx \] represents the area under the function \( y = \sqrt{64 - x^2} \) from \( x = -8 \) to \( x = 8 \).
This function describes the upper semicircle of a circle with radius 8, centered at the origin \((0, 0)\). The area under this curve represents half of the area of a full circle with radius 8.
### Calculation:
The area \(A\) of a full circle with radius \(r\) is given by the formula:
\[ A = \pi r^2 \]
For a circle with radius 8:
\[ A = \pi (8)^2 = 64\pi \]
Since the integral only considers the upper half of the circle (a semicircle), the area under the function is half of the total area:
\[ \text{Area under the curve} = \frac{1}{2} \times 64\pi = 32\pi \]
Therefore,
\[ \int_{-8}^{8} \sqrt{64 - x^2} \, dx = 32\pi \]
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