For Comrade Otto Bretscher

Calculus Level 5

Evaluate the flux integral , ∫ ∫ S G ⃗ . d S ⃗ \int \int_{S}\vec{G}.d\vec{S} where S S is the part of the surface z = 8 − x 2 − y z=8-x^{2}-y inside the cylinder , x 2 + y 2 = 1 x^{2}+y^{2}=1 oriented with normals pointing upward, and G ⃗ \vec{G} is the vector field , G ⃗ ( x , y , z ) = ⟨ 0 , 2 z , 2 y − 2 y 2 ⟩ \vec{G}(x,y,z)=\left \langle 0,2z,2y-2y^{2} \right \rangle

The answer is of the form A π A\pi submit the value of A A


The answer is 15.

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1 solution

Otto Bretscher
Mar 15, 2016

Ausgezeichnetes Problem!

∫ S G ⋅ d S = ∫ D ⟨ 0 , 16 − 2 x 2 − 2 y , 2 y − 2 y 2 ⟩ ⋅ ⟨ 2 x , 1 , 1 ⟩ d A = ∫ D ( 16 − 2 x 2 − 2 y 2 ) d A \int_{S}\mathbf{G}\cdot d\mathbf{S}=\int_{D}\langle0,16-2x^2-2y,2y-2y^2\rangle\cdot \langle2x,1,1\rangle dA=\int_{D}(16-2x^2-2y^2) dA = ∫ 0 2 π ∫ 0 1 ( 16 − 2 r 2 ) r d r d θ = 15 π =\int_{0}^{2\pi}\int_{0}^{1} (16-2r^2)r\enspace dr \enspace d\theta= \boxed{15}\pi , where D D is the disc x 2 + y 2 ≤ 1 x^2+y^2\leq 1

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