Use polar coordinates to compute R(x+y)dA where R is the region in the first quadrant between the circles of radius 1 and radius 3 centered at the origin.

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Question
### Advanced Calculus Problems in Multiple Coordinates Systems

#### Problem 6:
**Task:**
Use polar coordinates to compute the double integral 
\[ \iint_R (x+y) \, dA \]
where \( R \) is the region in the first quadrant between the circles of radius 1 and radius 3 centered at the origin.

**Explanation:** 
In this problem, you'll transform the given Cartesian coordinate integral into polar coordinates. The radii define the bounds: \(1 \leq r \leq 3\) and the first quadrant defines the angular range \(0 \leq \theta \leq \frac{\pi}{2}\).

#### Problem 7:
**Task:**
Calculate the triple integral 
\[ \iiint_E x^2 y^2 z^2 \, dV \]
where \( E = \{(x,y,z) \mid -1 \leq x \leq 1, -1 \leq y \leq 1, -1 \leq z \leq 1 \} \).

**Explanation:**
The integral is evaluated over a cuboid region in 3-dimensional space where \(x\), \(y\), and \(z\) all range from \(-1\) to 1.

#### Problem 8:
**Task:**
Compute the triple integral 
\[ \iiint_E z \sqrt{x^2 + y^2} \, dV \]
using spherical coordinates, where 
\[ E = \{(x,y,z) \mid x^2 + y^2 + z^2 \leq 1, x > 0, y > 0, z > 0 \} \].
That is, the first octant of the unit ball.

**Hints:**
1. Simplify \(x^2 + y^2\) after substitution of spherical variables.
2. The first octant determines the limits on the spherical coordinate \(\phi\) and \(\theta\).

**Explanation:**
The problem involves converting the integral from Cartesian to spherical coordinates and then evaluating it over the given bounds, which correspond to the first octant of a unit sphere. This simplifies to \(0 \leq \rho \leq 1\), \(0 \leq \phi \leq \frac{\pi}{2}\), and \(0 \leq \theta \leq \frac{\
Transcribed Image Text:### Advanced Calculus Problems in Multiple Coordinates Systems #### Problem 6: **Task:** Use polar coordinates to compute the double integral \[ \iint_R (x+y) \, dA \] where \( R \) is the region in the first quadrant between the circles of radius 1 and radius 3 centered at the origin. **Explanation:** In this problem, you'll transform the given Cartesian coordinate integral into polar coordinates. The radii define the bounds: \(1 \leq r \leq 3\) and the first quadrant defines the angular range \(0 \leq \theta \leq \frac{\pi}{2}\). #### Problem 7: **Task:** Calculate the triple integral \[ \iiint_E x^2 y^2 z^2 \, dV \] where \( E = \{(x,y,z) \mid -1 \leq x \leq 1, -1 \leq y \leq 1, -1 \leq z \leq 1 \} \). **Explanation:** The integral is evaluated over a cuboid region in 3-dimensional space where \(x\), \(y\), and \(z\) all range from \(-1\) to 1. #### Problem 8: **Task:** Compute the triple integral \[ \iiint_E z \sqrt{x^2 + y^2} \, dV \] using spherical coordinates, where \[ E = \{(x,y,z) \mid x^2 + y^2 + z^2 \leq 1, x > 0, y > 0, z > 0 \} \]. That is, the first octant of the unit ball. **Hints:** 1. Simplify \(x^2 + y^2\) after substitution of spherical variables. 2. The first octant determines the limits on the spherical coordinate \(\phi\) and \(\theta\). **Explanation:** The problem involves converting the integral from Cartesian to spherical coordinates and then evaluating it over the given bounds, which correspond to the first octant of a unit sphere. This simplifies to \(0 \leq \rho \leq 1\), \(0 \leq \phi \leq \frac{\pi}{2}\), and \(0 \leq \theta \leq \frac{\
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