- #1

Philip Wong

- 95

- 0

## Homework Statement

estimate the volume of the solid

z=-2(x^2+y^2)+8

between the two plates z=4 and z=0

## Homework Equations

In question like this, should I use triple integrals or double integrals in polar coordinates? I'm stuck in between which to use, because the question asks to estimate the volume which suggest a triple integrals. Yet the function gives me a strong feeling that I should use a double integrals in polar coordinates instead. Any suggestions?

Even though I'm not sure have I done it right, I tried solving it using double integrals in polar coordinates. Please check have I done it right, and if I should do it in triple integrals do please give me a guideline of how to do it (as it has not been taught to us yet, but the assignment is due before our next class)

p={(r,θ)= 0≤ r ≤4, 0≤ θ ≤ [itex]\pi[/itex]

∫∫[itex]_{p}[/itex] -2(x

^{2}+y

^{2})+8=∫[itex]^{\pi}_{0}[/itex]∫[itex]^{4}_{0}[/itex]

## The Attempt at a Solution

∫∫[itex]_{p}[/itex] -2(x

^{2}+y

^{2})+8=∫[itex]^{\pi}_{0}[/itex]∫[itex]^{4}_{0}[/itex]-2(r

^{2})r Δr Δθ

∫[itex]^{\pi}_{0}[/itex]∫[itex]^{4}_{0}[/itex] -2r

^{3}Δr Δθ

∫[itex]^{\pi}_{0}[/itex] [-2r

^{4}/4][itex]^{4}_{0}[/itex] Δθ

∫[itex]^{\pi}_{0}[/itex][-2(4)

^{4}/4]-[-2(0)

^{4}/4] Δθ

∫[itex]^{\pi}_{0}[/itex]-128 Δθ

[-128θ][itex]^{\pi}_{0}[/itex]

=-128[itex]\pi[/itex]