Area Between Curves

The Big Picture

= -

“Proof”

Find the limit of the area of rectangles given by a representative rectangle:

The area would be the limit of the sum of the areas of infinite representative rectangles with infinitely small intervals (n → ∞):

n

iii

nxxgxf

1

lim

This is the integral:

b

a

dxxgxf

Steps, Hints, & Tricks

If not given, find the points of intersection by setting equations equal to each other. These are your interval values (a and b)

Graphs may intersect in more than 2 points – find all that apply

If a graph is a function of y, then change your perspective horizontally (a and b would be y values)

If the graphs switch relative position (top-bottom or left-right), then you must break the integral into separate integrals (see above graph)

Example: Find the area between the curves

0 1

1

Example: Find the area between the curves

Area Between Curves

2

1

2

1

y

y

x

x

dyleftrightA

dxbottomtopA

Example: Find the area between the curves

2

0 4 -1