Consider the region in the –plane bounded above and below by and

MATHEMATICS · Multi-part · A Level · Mathematics · Integration · Integration

Problem

Consider the region in the

xy

–plane bounded above and below by

y=-2

and

y=3

, respectively. For every

y

in this interval the left boundary is the line

x=y-4

. The right boundary, however, is given piecewise by •

x=8-y^{2}

for

-2\le y\le a

•

x=2+y

for

a\le y\le 3

where

a

is the

y

–coordinate at which the two right–hand curves meet. (a) Solve for the value of

a

by finding the intersection of

x=8-y^{2}

and

x=2+y

. (b) Set up a sum of definite integrals whose value equals the exact area of the region. Do not combine the integrals into a single expression. (c) Evaluate the integrals from part (b) to obtain the exact area.

Related theory

Area Between Curves
Area Applications
Applications of Integration

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