Sec 6.1: Area Between 2 Curves
We have seen how to find the area between a curve and the x-axis. In
this section we will generalize that idea to find the area bounded by
two curves, f and g.
EXAMPLE: Sketch the region enclosed by the functions y = x + 2
and y = x2. Then set up and evaluate a definite integral to find the
exact area of the region.
Draw the picture.
Draw a representative
rectangle.
Example: find the area enclosed by y = x2 , y = x3, x = ­1 and x = 2
First, draw a picture of the area.
Draw the picture.
Draw a representative
rectangle.
We have always drawn vertical rectangles, where the width of the rectangle is
Δx. However, sometimes it is best to draw horizontal rectangles, with a width
of Δy. Then the limits of integration are values of y and we'll integrate with
respect to y.
Draw the picture.
2
Draw
a representative
EXAMPLE: Sketch the region enclosed by the functions x = 1 - y
rectangle.
and x = y2 - 1. Then set up and evaluate a definite integral to find
the exact area of the region.
EXAMPLE: Find the area between the graphs of y = sin 2x
and y = sin x between x = 0 and x = Π.
Draw the picture.
Draw a representative
rectangle.
You try some-Find the area between y = -sin x and y = -2sin x
1.
2.
Find the area between x = y^2 and y = -x.