4- x2 Change the Cartesian integral dy dx into an equivalent polar integral. Then evaluate the polar integral. -2J 0 Change the Cartesian integral into an equivalent polar integral. .2 4-x2 dy dx = |dr de -2 (Type exact answers, using t as needed.) The value of the double integral is. (Type an exact answer, using a as needed.)
4- x2 Change the Cartesian integral dy dx into an equivalent polar integral. Then evaluate the polar integral. -2J 0 Change the Cartesian integral into an equivalent polar integral. .2 4-x2 dy dx = |dr de -2 (Type exact answers, using t as needed.) The value of the double integral is. (Type an exact answer, using a 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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![**Changing the Cartesian Integral to a Polar Integral**
To solve the problem of changing the Cartesian integral into an equivalent polar integral and then evaluating the polar integral, follow these steps:
**Step-by-Step Instructions:**
1. **Given Cartesian Integral:**
\[ \int_{-2}^{2} \int_{0}^{\sqrt{4 - x^2}} dy \, dx \]
2. Change the Cartesian integral into an equivalent polar integral. The integral limits and the integrand need to be changed to polar coordinates (`r`, `θ`):
- \( x = r \cos(\theta) \)
- \( y = r \sin(\theta) \)
- \( dx \, dy = r \, dr \, d\theta \)
Substitute into the integral:
\[
\int_{-2}^{2} \int_{0}^{\sqrt{4 - x^2}} dy \, dx = \int_{0}^{2\pi} \int_{0}^{2} r \, dr \, d\theta
\]
3. Evaluate the polar integral:
- Integration with respect to \( r \):
\[
\int_{0}^{2} r \, dr = \left[ \frac{r^2}{2} \right]_{0}^{2} = \frac{4}{2} - \frac{0}{2} = 2
\]
- Integration with respect to \( \theta \):
\[
\int_{0}^{2\pi} d\theta = [\theta]_{0}^{2\pi} = 2\pi - 0 = 2\pi
\]
Multiply the results of these integrals:
\[
2 \times 2\pi = 4\pi
\]
Therefore:
**The value of the double integral is:**
\[4\pi\]
**Instruction:**
Enter your answer for each component and the final value obtained into the provided answer boxes on the educational website.
---
**Formulas and Conversion Details:**
1. The provided Cartesian integral is quickly recognizable as describing the area of a semicircle in two-dimensional space.
2. Polar coordinates facilitate the integral solution by transforming §dx/dy boundaries to circular domains, simplifying the integral calculations.
---
**Final Answer:**
- \(\int_0^{2\](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1bedffc2-fb9b-46af-a153-6d9ee06b6f91%2Fa519834a-14b9-4553-a816-9914b9a2adbe%2F6x5czc_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Changing the Cartesian Integral to a Polar Integral**
To solve the problem of changing the Cartesian integral into an equivalent polar integral and then evaluating the polar integral, follow these steps:
**Step-by-Step Instructions:**
1. **Given Cartesian Integral:**
\[ \int_{-2}^{2} \int_{0}^{\sqrt{4 - x^2}} dy \, dx \]
2. Change the Cartesian integral into an equivalent polar integral. The integral limits and the integrand need to be changed to polar coordinates (`r`, `θ`):
- \( x = r \cos(\theta) \)
- \( y = r \sin(\theta) \)
- \( dx \, dy = r \, dr \, d\theta \)
Substitute into the integral:
\[
\int_{-2}^{2} \int_{0}^{\sqrt{4 - x^2}} dy \, dx = \int_{0}^{2\pi} \int_{0}^{2} r \, dr \, d\theta
\]
3. Evaluate the polar integral:
- Integration with respect to \( r \):
\[
\int_{0}^{2} r \, dr = \left[ \frac{r^2}{2} \right]_{0}^{2} = \frac{4}{2} - \frac{0}{2} = 2
\]
- Integration with respect to \( \theta \):
\[
\int_{0}^{2\pi} d\theta = [\theta]_{0}^{2\pi} = 2\pi - 0 = 2\pi
\]
Multiply the results of these integrals:
\[
2 \times 2\pi = 4\pi
\]
Therefore:
**The value of the double integral is:**
\[4\pi\]
**Instruction:**
Enter your answer for each component and the final value obtained into the provided answer boxes on the educational website.
---
**Formulas and Conversion Details:**
1. The provided Cartesian integral is quickly recognizable as describing the area of a semicircle in two-dimensional space.
2. Polar coordinates facilitate the integral solution by transforming §dx/dy boundaries to circular domains, simplifying the integral calculations.
---
**Final Answer:**
- \(\int_0^{2\
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