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Let R be the region given by 0 SzS a? + y? , a² + y? <3. Evaluate (x + 2)dV.

Let R be the region given by 0 SzS a? + y? , a² + y? <3. Evaluate (x + 2)dV.

Advanced Engineering Mathematics
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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Problem. Let R be the region given by 0 < z < x² + y² , x² + y? < 3. Evaluate (x + 2)dV. Explanation. This is the region under a paraboloid and inside a cylinder. The reason cylindrical coordinates would be a good coordinate system to pick is that the condition x² + y? < 3 means we will probably go to polar later anyway, so we can just go there now with cylindrical coordinates. The paraboloid's equation in cylindrical coordinates (i.e. in terms of r, 0, and z) is ? z = x + y? Thus, our bounds for z will be くzく Now that we have z, we can look at the cy-plane for our polar bounds. The disc x² + y² < 3 in polar is <rs Therefore, the integral becomes (x + 2)dV = II, f(r(r, 0, 2)rdz dr de dz dr de
Transcribed Image Text:Problem. Let R be the region given by 0 < z < x² + y² , x² + y? < 3. Evaluate (x + 2)dV. Explanation. This is the region under a paraboloid and inside a cylinder. The reason cylindrical coordinates would be a good coordinate system to pick is that the condition x² + y? < 3 means we will probably go to polar later anyway, so we can just go there now with cylindrical coordinates. The paraboloid's equation in cylindrical coordinates (i.e. in terms of r, 0, and z) is ? z = x + y? Thus, our bounds for z will be くzく Now that we have z, we can look at the cy-plane for our polar bounds. The disc x² + y² < 3 in polar is <rs Therefore, the integral becomes (x + 2)dV = II, f(r(r, 0, 2)rdz dr de dz dr de
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