Write a triple integral in cylindrical coordinates of the function f(x, y) = x+y over the region bounded by the xy-plane, z = 20 - 3x² - 3y2, and z= x² + y².

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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Write a triple integral in cylindrical coordinates of the function \( f(x,y) = x + y \) over the region bounded by the xy-plane, \( z = 20 - 3x^2 - 3y^2 \), and \( z = x^2 + y^2 \).
Transcribed Image Text:Write a triple integral in cylindrical coordinates of the function \( f(x,y) = x + y \) over the region bounded by the xy-plane, \( z = 20 - 3x^2 - 3y^2 \), and \( z = x^2 + y^2 \).
Expert Solution
Step 1

 A triple integral involves the integral with respect to the product of variables x,y and z. In this problem, we have to set up a triple integral in the cylindrical coordinates of the function fx,y=x+y, over the region bounded by xy plane, z=20-3x2-3y2 and z=x2+y2. The equation connecting cartesian coordinates x,y,z and cylindrical coordinates r,θ,z is given by x=rcosθ, y=rsinθ and z=z.

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