Find the volume of the solid enclosed by the paraboloids z = 25( x^2 + y^2 ) and z = 50 - 25( x^2 + y^2).
I get to a point where I set the double integral to 50-50(x^2+y^2)dA, but what I am having trouble with is finding the region to evaluate this problem from. I have another problem like this and it doesn't make sense either.
I get to a point where I set the double integral to 50-50(x^2+y^2)dA, but what I am having trouble with is finding the region to evaluate this problem from. I have another problem like this and it doesn't make sense either.
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Note that the curve of intersection is the unit circle, because
25(x^2 + y^2) = 50 - 25(x^2 + y^2) ==> x^2 + y^2 = 1.
So, using polar coordinates yields
∫∫ (50 - 50(x^2 + y^2)) dA
= ∫(θ = 0 to 2π) ∫(r = 0 to 1) 50(1 - r^2) * r dr dθ
= 100π ∫(r = 0 to 1) (r - r^3) dr
= 100π (1/2 - 1/4)
= 25π.
I hope this helps!
25(x^2 + y^2) = 50 - 25(x^2 + y^2) ==> x^2 + y^2 = 1.
So, using polar coordinates yields
∫∫ (50 - 50(x^2 + y^2)) dA
= ∫(θ = 0 to 2π) ∫(r = 0 to 1) 50(1 - r^2) * r dr dθ
= 100π ∫(r = 0 to 1) (r - r^3) dr
= 100π (1/2 - 1/4)
= 25π.
I hope this helps!