Find the volume between z=2x^2+3y^2 and z=4. Is there a way finding that with a double integral?

Matonya

Matonya

Answered question

2022-08-12

Volume between under parabaloid
Suppose I want to find the volume between z = 2 x 2 + 3 y 2 and z = 4. Is there a way finding that with a double integral? I tried to use 4 2 x 2 3 y 2 inside the integral and then convert it to polar coordinates...

Answer & Explanation

Dominic Paul

Dominic Paul

Beginner2022-08-13Added 17 answers

Step 1
The volume of the solid of interest E is,
D ( 4 2 x 2 3 y 2 ) d A
Where D is the projection of the solid E onto the xy plane. That is, the region 2 x 2 + 3 y 2 4 in the xy plane. Or, ( x 2 ) 2 + ( y 4 3 ) 2 1.
To evaluate the double integral, use the modified polar transformation:
x = 2 r cos ( θ )
y = 4 3 r sin ( θ )
Whose Jacobian is 8 3 r. Hence the volume is, V = 0 2 π 0 1 8 3 r ( 4 4 r 2 ) d r d θ
= 4 π 2 3
= 4 3 π 6
The above transformation can be thought of as combining two transformations.
X = x 2
Y = y 4 3
Then X = r cos ( θ )
Y = r sin ( θ )
Step 2
At a height of z 0, a slice of our solid of interest (parallel to the xy plane) has elliptical area A ( z ) = π z 2 z 3 = π 6 z.
So the volume of the solid is, V = 0 4 A ( z ) d z
= 8 π 6
= 4 3 π 6
Which matches the result from the double integral.

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