Question

Integrate the given function over the given surface. $G(x,y,z) = 2z^2$, over the hemisphere $x^2 + y^2 + z^2 = 4$, $z \ge 0$ The value of the surface integral is $\boxed{}$. (Type an exact answer, using $\pi$ as needed.)

Name: integrate the given function over the given surface gxyz 2z2 over the hemisphere x2 y2 z2 4 z ge 0 the value of the surface integral is boxed type an exact answer using pi as needed 87287
Uploaded: 2024-12-01T23:05:05-08:00
Duration: 2 min 10 s
Channel: Nick Johnson
Description: integrate the given function over the given surface gxyz 2z2 over the hemisphere x2 y2 z2 4 z ge 0 the value of the surface integral is boxed type an exact answer using pi as needed 87287

          Integrate the given function over the given surface.
$G(x,y,z) = 2z^2$, over the hemisphere $x^2 + y^2 + z^2 = 4$, $z \ge 0$
The value of the surface integral is $\boxed{}$.
(Type an exact answer, using $\pi$ as needed.)

integrate the given function over the given surface gxyz 2z2 over the hemisphere x2 y2 z2 4 z ge 0 the value of the surface integral is boxed type an exact answer using pi as needed 87287

Added by Shawn L.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Nick Johnson

12/01/2024

Step 1

We want to compute the surface integral $\iint_S G(x,y,z) dS = \iint_S 2z^2 dS$. We use spherical coordinates: $x = 2\sin\phi\cos\theta$ $y = 2\sin\phi\sin\theta$ $z = 2\cos\phi$ where $0 \le \phi \le \frac{\pi}{2}$ and $0 \le \theta \le 2\pi$. Then $dS = Show more…

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Integrate the given function over the given surface. $G(x,y,z) = 2z^2$, over the hemisphere $x^2 + y^2 + z^2 = 4$, $z \ge 0$ The value of the surface integral is $\boxed{}$. (Type an exact answer, using $\pi$ as needed.)