Double Integration using Cartesian Coordinates

Example: Find the area bounded by the curves y = x² and y = x

in the first quadrant See the figure below.

Strategy: Draw the element of area (very important) and determine the points of

intersection which will provide the limits of integration.

The curve y = x² lies below y = x. The points of intersection are determined

by

x = x² So x = 0 and x = 1

Now the element of area is dA where dA = dx dy = dy dx

For the element shown above, choose to integrate on the variable y first.

i.e. Sweep in the “y-direction first. So dA = dy dx and the integral becomes:

x =1 y = x x =1 x

A = ∫ ∫ dy dx = ∫ y | dx

x = 0 y = x² x = 0 x²

x =1 1

A = ∫ [ x - x² ] dx = [ (1/2) x² - (1/3) x³ ] | = 1/6

x = 0 0

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