Chapter 14.5, Problem 27E

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347

Chapter
Section

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347
Textbook Problem

# Setting Up a Double IntegralIn Exercises 27–30, set up a double integral that represents the area of the surface given by z = f ( x , y ) that lies above the region R . f ( x , y ) = e x y ,       R = { ( x , y ) : 0 ≤ x ≤ 4 ,     0 ≤ y ≤ 10 }

To determine

To calculate: A double integral that represents the surface area of f(x,y)=exyz=f(x,y) lying above the region R R={(x,y): 0x4, 0y10}

Explanation

Given:

The surface is f(x,y)=exy lying above the region R represented by,

R={(x,y):Â 0â‰¤xâ‰¤4,Â 0â‰¤yâ‰¤10}.

Formula used:

The surface area can be calculated of the region R by,

S=âˆ¬R1+[fx(x,y)]2+[fy(x,y)]2dA

Calculation:

The surface f(x,y)=exy above the region R={(x,y):Â 0â‰¤xâ‰¤4,Â 0â‰¤yâ‰¤10}.

First, with respect to x, find the partial differentiation of the surface with the use of

ddx(xn)=nxnâˆ’1,ddx(constant)=0,ddx(ex)=ex.

fx(x,y)=ddx(exy)=yexy

Now, with respect to y

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