INTRODUCTION TO CALCULUS
MATH 1A
Unit 20: Area
Lecture
Rb
20.1. If f (x) ≥ 0, then a f (x) dx is the area under the graph of f (x) and above
Rb
the interval [a, b] on the x axes. If the function is negative, then a f (x) dx is negative
too and the integral is minus the area below the curve:
Rb
Therefore, a f (x) dx is the difference of the area above the graph minus the area
below the graph.
20.2. More generally we can also look at areas sandwiched between two graphs
f and g.
The area of a region G enclosed by two graphs f ≤ g and bound by a ≤ x ≤ b is
Z b
g(x) − f (x) dx
a
.
20.3. Make sure that if you have to compute such an integral that g ≥ f before giving
it the interpretation of an area.
Example: Find the area of the region enclosed by the x-axes, the y-axes and the
R π/2
graph of the cos function. Solution: 0 cos(x) dx = 1.
Example: Find the area of the region enclosed by the graphs f (x) = x2 and f (x) =
x4 .
MATH 1A
Unit 20: Area
Lecture
Rb
20.1. If f (x) ≥ 0, then a f (x) dx is the area under the graph of f (x) and above
Rb
the interval [a, b] on the x axes. If the function is negative, then a f (x) dx is negative
too and the integral is minus the area below the curve:
Rb
Therefore, a f (x) dx is the difference of the area above the graph minus the area
below the graph.
20.2. More generally we can also look at areas sandwiched between two graphs
f and g.
The area of a region G enclosed by two graphs f ≤ g and bound by a ≤ x ≤ b is
Z b
g(x) − f (x) dx
a
.
20.3. Make sure that if you have to compute such an integral that g ≥ f before giving
it the interpretation of an area.
Example: Find the area of the region enclosed by the x-axes, the y-axes and the
R π/2
graph of the cos function. Solution: 0 cos(x) dx = 1.
Example: Find the area of the region enclosed by the graphs f (x) = x2 and f (x) =
x4 .
