Calculate this triple integral over D: \int\int\int\sqrt{10+y^2

lenlifiauw2

lenlifiauw2

Answered question

2021-11-13

Calculate this triple integral over D:
10+y2x2dxdydz
over the region bounded D:
D={(x,y,z)x2+z2y2+10, x2+y29}

Answer & Explanation

Uersfeldte

Uersfeldte

Beginner2021-11-14Added 20 answers

First of all we see from x2+y2=9 that y lives in Jy=[3,3]
Now we can compute using Fubini:
D10+y2x2dxdydz
=y[3,3]dyxJx(y)dxzJz(x,y)10+y2x2
=y[3,3]dyxJx(y)dx210+y2x210+y2x2
=2y[3,3]dyxJx(y)dx(10+y2x2)
=2y[3,3]dy[10x+xy213x3]x=9y2x=+9+y2
=4y[3,3]dy9y2(10+y213(9y2))
=4339y2(7+43y2)dy
Substitution y=3sint
=4π2π23cost(7+439sin2t)3costdt

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