Use a double integral in polar coordinates to find the volume V of the solid bounded by the graphs of the equations. √x² + y² V = Z = √x Z = 0 x² + y² = 4 NR X 2 √x² +1² X dr de =

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
icon
Related questions
Question
**Transcription for Educational Website**

**Use a double integral in polar coordinates to find the volume \( V \) of the solid bounded by the graphs of the equations.**

Equations:
\[ z = \sqrt{x^2 + y^2} \]
\[ z = 0 \]
\[ x^2 + y^2 = 4 \]

Integral for Volume:
\[ V = \int_0^{2\pi} \int_0^2 \sqrt{x^2 + y^2} \, dr \, d\theta \]

**Explanation:**
- The problem involves finding the volume of a solid using integration in polar coordinates.
- The equations given describe a surface, a plane, and a circle.
- The correct setup for the double integral is:
  - The outer integral limits for \( \theta \) are from 0 to \( 2\pi \), representing a full rotation around the circle.
  - The inner integral limits for \( r \) are from 0 to 2, covering the radius from the center to the edge of the circle in the \( xy \)-plane.
- The integral involves calculating over the region defined by the circle \( x^2 + y^2 = 4 \).
- \( \sqrt{x^2 + y^2} \) is the function to be integrated, corresponding to the height \( z \) of the solid at each point in the circular region.
Transcribed Image Text:**Transcription for Educational Website** **Use a double integral in polar coordinates to find the volume \( V \) of the solid bounded by the graphs of the equations.** Equations: \[ z = \sqrt{x^2 + y^2} \] \[ z = 0 \] \[ x^2 + y^2 = 4 \] Integral for Volume: \[ V = \int_0^{2\pi} \int_0^2 \sqrt{x^2 + y^2} \, dr \, d\theta \] **Explanation:** - The problem involves finding the volume of a solid using integration in polar coordinates. - The equations given describe a surface, a plane, and a circle. - The correct setup for the double integral is: - The outer integral limits for \( \theta \) are from 0 to \( 2\pi \), representing a full rotation around the circle. - The inner integral limits for \( r \) are from 0 to 2, covering the radius from the center to the edge of the circle in the \( xy \)-plane. - The integral involves calculating over the region defined by the circle \( x^2 + y^2 = 4 \). - \( \sqrt{x^2 + y^2} \) is the function to be integrated, corresponding to the height \( z \) of the solid at each point in the circular region.
Expert Solution
steps

Step by step

Solved in 3 steps with 3 images

Blurred answer
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
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…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,