   Chapter 16.7, Problem 11E

Chapter
Section
Textbook Problem

Evaluate the surface integral.11. ∫∫s x dS, S is the triangular region with vertices (1, 0, 0), (0, -2, 0), and (0, 0, 4)

To determine

To find: The value of SxdS .

Explanation

Given data:

Triangular region vertices are (1,0,0),(0,2,0) and (0,0,4) .

Formula used:

Sf(x,y,z)dS=Df(x,y,g(x,y))(zx)2+(zy)2+1dA (1)

The equation of the plane through the points (1,0,0),(0,2,0) and (0,0,4) is 4x2y+z=4 .

Rearrange equation.

z=44x+2y (2)

Find the x limits by considering y and z as 0 in equation (1).

(0)=44x+2(0)4x=4x=44x=1

Find the y limits by considering z as 0 in equation (1).

0=44x+2y2y=4+4x2y=2(2x2)y=2x2

The regions of D is 0x1 and 2x2y0 .

Find zx .

Take partial differentiation for equation (2) with respect to x.

zx=x(44x+2y)=x(4)4x(x)+2yx(1)=(0)4(1)+2y(0)=4

Find zy .

Take partial differentiation for equation (2) with respect to y.

zy=y(44x+2y)=y(4)4xy(1)+2y(y)=(0)2x(0)+2(1)=2

Find SxdS .

Modify equation (1) as follows

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