Graph the integrand, and use area to evaluate the definite integral √ √64-x² dx. -8 8 ja √ √64-x² dx, as determined by the area under the graph of the integral, is -8 (Type an exact answer, using as needed.) The value of the definite integral
Graph the integrand, and use area to evaluate the definite integral √ √64-x² dx. -8 8 ja √ √64-x² dx, as determined by the area under the graph of the integral, is -8 (Type an exact answer, using as needed.) The value of the definite integral
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
![### Calculating the Definite Integral using Area
#### Problem Statement
Graph the integrand, and use area to evaluate the definite integral:
\[
\int_{-8}^{8} \sqrt{64 - x^2} \, dx
\]
#### Solution
To solve this problem, we need to determine the area under the curve of the given function from \(x = -8\) to \(x = 8\).
**Function:**
The integrand \(\sqrt{64 - x^2}\) represents the equation of a semicircle centered at the origin with a radius of 8.
**Approach:**
1. **Graphing the Function:**
- The function \(\sqrt{64 - x^2}\) is equivalent to the upper half of the circle with equation \(x^2 + y^2 = 64\).
- This circle has a center at \((0, 0)\) and a radius of 8.
- Only the upper half of the circle is considered due to the square root, resulting in a semicircular region above the x-axis.
2. **Calculating the Area:**
- The full area of a circle with radius 8 is \( \pi \times 8^2 = 64\pi \).
- Therefore, the area of the semicircle, which represents the area under the curve from \(x = -8\) to \(x = 8\), is half of the total circle area: \( 32\pi \).
Thus, the value of the definite integral \(\int_{-8}^{8} \sqrt{64 - x^2} \, dx\) is \( \boxed{32\pi} \).
*(Type an exact answer, using \(\pi\) as needed.)*](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F93ade889-6aa6-440d-b6d3-23ce5a98ca6e%2Fe2664909-8ab5-47d7-87fe-2fa27601d604%2Fxz7b6zl_processed.png&w=3840&q=75)
Transcribed Image Text:### Calculating the Definite Integral using Area
#### Problem Statement
Graph the integrand, and use area to evaluate the definite integral:
\[
\int_{-8}^{8} \sqrt{64 - x^2} \, dx
\]
#### Solution
To solve this problem, we need to determine the area under the curve of the given function from \(x = -8\) to \(x = 8\).
**Function:**
The integrand \(\sqrt{64 - x^2}\) represents the equation of a semicircle centered at the origin with a radius of 8.
**Approach:**
1. **Graphing the Function:**
- The function \(\sqrt{64 - x^2}\) is equivalent to the upper half of the circle with equation \(x^2 + y^2 = 64\).
- This circle has a center at \((0, 0)\) and a radius of 8.
- Only the upper half of the circle is considered due to the square root, resulting in a semicircular region above the x-axis.
2. **Calculating the Area:**
- The full area of a circle with radius 8 is \( \pi \times 8^2 = 64\pi \).
- Therefore, the area of the semicircle, which represents the area under the curve from \(x = -8\) to \(x = 8\), is half of the total circle area: \( 32\pi \).
Thus, the value of the definite integral \(\int_{-8}^{8} \sqrt{64 - x^2} \, dx\) is \( \boxed{32\pi} \).
*(Type an exact answer, using \(\pi\) as needed.)*
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