Finding the volume of an elliptical cone with double integralsFor my class I need to find the volume of an elliptical cone bounded by z = 9 x 2 + y 2 and the plane z = 2. My thought process was to integrate the equation for z = 9 x 2 + y 2 over the region bounded by the projection onto the xy-plane. This is the integral I set up: ∫ − 2 2 ∫ − 1 3 4 − y 2 1 3 4 − y 2 9 x 2 + y 2 d x d yTo check my answer, I looked up and found that the volume of an elliptical cone can be found using the equation: V = 1 3 π a b hWhen I checked my answer I got from the double integral, I found that it is 4 times what it should be. Can anyone explain to me what I did wrong?

Question

Finding the volume of an elliptical cone with double integralsFor my class I need to find the volume of an elliptical cone bounded by   z  =      9          x      2        +          y      2       and the plane   z  =  2. My thought process was to integrate the equation for   z  =      9          x      2        +          y      2       over the region bounded by the projection onto the xy-plane. This is the integral I set up:      ∫          −      2        2        ∫          −              1        3                    4        −                  y          2                                    1        3                    4        −                  y          2                          9          x      2        +          y      2          d  x    d  yTo check my answer, I looked up and found that the volume of an elliptical cone can be found using the equation:  V  =      1    3    π  a  b  hWhen I checked my answer I got from the double integral, I found that it is 4 times what it should be. Can anyone explain to me what I did wrong?

Jamir Young · Accepted Answer

Step 1The volume integral should be,  V  =      ∫          −      2        2        ∫          −              1        3                    4        −                  y          2                                    1        3                    4        −                  y          2                          (    2    −          9              x        2            +              y        2              )      d  x    d  yStep 2Note that the volume specified in the problem is between the elliptic cone the plane   z  =  2. Your integral expression is for the volume between the cone and the plane   z  =  0

Find the volume of an elliptical cone bounded by z=sqrt{9x^2+y^2} and the plane z=2. My thought process was to integrate the equation for z=sqrt{9x^2+y^2} over the region bounded by the projection onto the xy-plane.

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