$Evaluate the surface integral $\iint_{S}F \cdot dn$ over the given surface $S$$

Question

$Evaluate the surface integral $\iint_{S}F \cdot dn$ over the given surface $S$$

Evaluate $\iint_{S}F \cdot dn$ where $F=(x^2,0,3y^2)$ and $S$ is the potion of the plane $x+y+z=1$ in the first octant.

Multivariable Calculus Integrals Stokes' Theorem

Elvin

21

Report

Answer

The answer is accepted.

Join Matchmaticians Affiliate Marketing Program to earn up to a 50% commission on every question that your affiliated users ask or answer.

Answer 1

Answers can only be viewed under the following conditions:

The questioner was satisfied with and accepted the answer, or
The answer was evaluated as being 100% correct by the judge.

View the answer

The surface $S$ projected to $xy$ plane is $T=\{(x,y): 0\leq x \leq 1, 0\leq y \leq 1-x\}.$ Also the normal to the surface $S$ with $z=1-x-y$ is $n=(1,1,1)/\sqrt{3}$. note that the surface $S$ is given by $g(x,y,z)=x+y+z=1$, and $|\nabla g|=|(1,1,1,)|=\sqrt{3}$.

So
\[\iint_S F\cdot n=\iint F \cdot n \sqrt{3}dx dy=\iint_T x^2+3y^2 dx dy=\int_0^1\int_0^{1-x}x^2+3y^2dydx\]
\[=\int_0^1 (x^2y+y^3)|_0^{1-x}dx=\int_0^1 x^2(1-x)+(1-x)^3=\int_0^1 (1-3x+4x^2-2x^3)=\frac{1}{3}.\]

Erdos

4.7K

$Evaluate the surface integral $\iint_{S}F \cdot dn$ over the given surface $S$$

Answer

Related Questions

Search