   Chapter 15.5, Problem 23E

Chapter
Section
Textbook Problem

Find the area of the finite part of the paraboloid y = x2 + z2 cut off by the plane y = 25. [Hint: Project the surface onto the xz-plane.]

To determine

To find: The area of the given region.

Explanation

Formula used:

The surface area with equation z=f(x,y),(x,y)D , where fx and fy are continuous, is A(S)=D[fx(x,y)]2+[fy(x,y)]2+1dA .

Here, D is the given region.

If f is a polar rectangle R given by 0arb,αθβ, where 0βα2π , then, Rf(x,y)dA=αβabf(rcosθ,rsinθ)rdrdθ (1)

If g(x) is the function of x and h(y) is the function of y then,

abcdg(x)h(y)dydx=abg(x)dxcdh(y)dy (2)

Given:

The finite part of the paraboloid, y=x2+z2 cutoff by the plane y=25 .

Calculation:

From the hint given in the problem, project the region onto the xz-plane that is substitute y=25 in y=x2+z2 . This yields, x2+z2=25 which means in xz-plane, the region is the circle of radius 5. So, use polar coordinates to evaluate the area of the region.

The partial derivatives fx and fz are,

fx=2xfz=2z

Then, by the formula mentioned above, the area of the surface is given by,

A(S)=D(2x)2+(2z)2+1dA=D4x2+4z2+1dA=D4(x2+z2)+1dA

Since the equation involves only x and z, substitute x=rcosθ,z=sinθ

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