Sketch the region whose area is given by the definite integral. Then use a geometric formula to evaluate the integral (a > 0, r > 0). 1₁√9-x²0x dx Step 1 Sketch the region whose area is given by the definite integral. Step 2 x² dx -3 -2 -1 y 6 5 4 2 1 1 Note that the region given by the definite integral 2 Livo 3 X 9x² dx is a semicircle of radius
Sketch the region whose area is given by the definite integral. Then use a geometric formula to evaluate the integral (a > 0, r > 0). 1₁√9-x²0x dx Step 1 Sketch the region whose area is given by the definite integral. Step 2 x² dx -3 -2 -1 y 6 5 4 2 1 1 Note that the region given by the definite integral 2 Livo 3 X 9x² dx is a semicircle of radius
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Transcribed Image Text:Sketch the region whose area is given by the definite integral. Then use a geometric formula to evaluate the integral (a > 0, r > 0).
3
[²√9-x² dx
Step 1
Sketch the region whose area is given by the definite integral.
Step 2
13
3
9 - x² dx
-3 -2
7
-1
y
Submit Skip (you cannot come back)
5
4
2
Note that the region given by the definite integral
1 2 3
3
Bvo-x2a
X
9 - x² dx is a semicircle of radius
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 4 steps with 4 images
Follow-up Questions
Read through expert solutions to related follow-up questions below.
Follow-up Question
![**Educational Explanation**
To determine the area of a region using calculus, consider the following steps:
### Integral Problem Statement
- The goal is to evaluate the definite integral:
\[
\int_{-3}^{3} \sqrt{9 - x^2} \, dx
\]
### Step 1: Sketch the Region
- The integral represents the area under the curve given by \( y = \sqrt{9 - x^2} \).
- Sketch this curve as the top half of a circle centered at the origin with radius 3.
- The shaded region depicts a semicircle from \( x = -3 \) to \( x = 3 \).
### Graph Explanation
- The graph shows a semicircle with a radius of 3, spanning horizontally from \(-3\) to \(3\) along the x-axis.
- The y-axis ranges from 0 to 3, representing the height of the semicircle.
### Step 2: Interpretation
- Recognize that the region specified by the integral \(\int_{-3}^{3} \sqrt{9 - x^2} \, dx\) is a semicircle with radius 3.
### Step 3: Calculation Using Geometry
- To find the area of the semicircle, use the formula:
\[
\text{Area of a semicircle} = \frac{1}{2} \pi r^2
\]
- Here, \( r = 3 \).
- Complete the formula to find the area of the semicircle:
\[
\frac{1}{2} \pi \times 3^2
\]
### Conclusion
Submit your answer to verify the area calculation is correct. This involves basic integration and geometric understanding of circles.](https://content.bartleby.com/qna-images/question/9ba58da1-aa6c-4b34-ac65-53856b97f0bf/7be18faf-96df-44ef-adc5-00fc25a8a66f/2d4yqda_processed.png)
Transcribed Image Text:**Educational Explanation**
To determine the area of a region using calculus, consider the following steps:
### Integral Problem Statement
- The goal is to evaluate the definite integral:
\[
\int_{-3}^{3} \sqrt{9 - x^2} \, dx
\]
### Step 1: Sketch the Region
- The integral represents the area under the curve given by \( y = \sqrt{9 - x^2} \).
- Sketch this curve as the top half of a circle centered at the origin with radius 3.
- The shaded region depicts a semicircle from \( x = -3 \) to \( x = 3 \).
### Graph Explanation
- The graph shows a semicircle with a radius of 3, spanning horizontally from \(-3\) to \(3\) along the x-axis.
- The y-axis ranges from 0 to 3, representing the height of the semicircle.
### Step 2: Interpretation
- Recognize that the region specified by the integral \(\int_{-3}^{3} \sqrt{9 - x^2} \, dx\) is a semicircle with radius 3.
### Step 3: Calculation Using Geometry
- To find the area of the semicircle, use the formula:
\[
\text{Area of a semicircle} = \frac{1}{2} \pi r^2
\]
- Here, \( r = 3 \).
- Complete the formula to find the area of the semicircle:
\[
\frac{1}{2} \pi \times 3^2
\]
### Conclusion
Submit your answer to verify the area calculation is correct. This involves basic integration and geometric understanding of circles.
Solution
by Bartleby ExpertRecommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
