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Chapter 13: Q 16. (page 1014)

Let h1(y)andh2(y)be two continuous functions such that g1(x)鈮�g2(x)on[a,b]and let 惟be region in xyplane bounded by g1andg2on[a,b]. Use your answer to Exercise 14 to set up an iterated integral whose value is the area of 惟. How is this iterated integral related

to the definite integral you would have used to compute the area of 惟in Chapter 4?

Short Answer

Expert verified

The iterated integral that gives area of region is鈭�cdh2(y)鈭�h2dxdy

Step by step solution

01

Given Information

Two continuous functions h1(y)andh2(y)are given such that h1(y)<h2(y), in c,d

Assume惟be region inxyplane bounded by curvesh1andh2inc,d

02

Simplification

Consider arbitrary type II region.

The integral 鈭�惟dAtype type II is calculated by taking elemental area of width 螖ywith left end lying over x=h1(y)and right end lying over x=h2(y)

Surface integral becomes

鈭�惟dA=鈭�ch1(y)dh2(y)dxdy

鈭�惟dA=鈭�cd[x]h1(y)h2(y)dy

=鈭�cdh2(y)-h1(y)dy

Hence, the iterated integral that gives area of region is鈭�cdh2(y)鈭�1h2(y)dxdy

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Most popular questions from this chapter

Let 惟be a lamina in the xy-plane. Suppose 惟is composed of two non-overlapping lamin 惟1and 惟2, as follows:

Show that if the masses and centers of masses of 惟1and 惟2are m1and m2,and x炉1,y炉1andx炉2,y炉2respectively, then the center of mass of 惟is x炉,y炉,where

x炉=m1x炉1+m2x炉2m1+m2andy炉=m1y炉1+m2y炉2m1+m2

Evaluate each of the double integral in the exercise 37-54 as iterated integrals

鈭�鈭�R(x-ey)dAWhereR={(x,y)|-3鈮�x鈮�2,-2鈮�y鈮�2}

Evaluate the sums in Exercises 23鈥�28.

鈭�i=14鈭�j=13鈭�k=12ij2k3

In the following lamina, all angles are right angles and the density is constant:

Explain how the Fundamental Theorem of Calculus is used in evaluating the iterated integral鈭�cd鈭�abfx,ydxdy.
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