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Chapter 14: Q. 23 (page 1119)

S is the portion of the saddle surface determined by z = x2 鈭� y2 that lies above and/or below the annulus in the xy-plane determined by the circles with radii

32and2

and centered at the origin.

Short Answer

Expert verified

Hence the area is19蟺6

Step by step solution

01

Step 1. Given 

The given equation isz = x2 鈭� y2.

02

Step 2.  Formula for finding the area of surface

IfasurfaceSisgivenbyz=f(x,y)forf(x,y)鈭�D鈯�R2Thenthesurfaceareaofsmoothsurfaceis鈭�S鈥�dS=鈭�S鈥�鈭�z鈭�x2+鈭�z鈭�y2+1dA=鈭�D鈥�鈭�z鈭�x2+鈭�z鈭�y2+1dA鈰�鈰�(1

03

Step 3.  Finding the partial derivative 

x=y+zthen,z=x2-y2now,firstfind鈭�z鈭�x=鈭�鈭�xx2-y2=2xand鈭�z鈭�y=鈭�鈭�yx2-y2=-2y

04

Step 4. Graph

viewed it as a x-simple, then the region of integration will be

D=r,胃|32鈮�r鈮�2,0鈮�胃鈮�2蟺

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Most popular questions from this chapter

If S is parametrized by r(u, v), why is dS=ru脳rvthe correct factor to use to account for distortion of area?

Do the vectors in the range of F(x,y)=xi+yjpoint towards or away from the origin?

Make a chart of all the new notation, definitions, and theorems in this section, including what each new thing means in terms you already understand.

Find the area of S is the portion of the plane with equation x = y + z that lies above the region in the xy-plane that is bounded by y = x, y = 5, y = 10, and the y-axis.

Evaluate the integrals in Exercises 43鈥�46 directly or using Green鈥檚 Theorem.

鈭�R3xy-4x2ydA, where R is the unit disk.
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