Areas by Integration
1. Area under a curve – region bounded by the given function, vertical lines
and the x –axis.
2. Area under a curve – region bounded by the given function, horizontal
lines and the y –axis.
3. Area between curves defined by two given functions.
1. Area under a curve – region bounded by the given function, vertical
lines and the x –axis.
If f(x) is a continuous and nonnegative function of x on the closed interval [a, b], then the
area of the region bounded by the graph of f, the x-axis and the vertical lines x=a and x=b
is given by:
b
Area f ( x)dx
a
A
When calculating the area under a curve f(x), follow the steps below:
1. Sketch the area.
2. Determine the boundaries a and b,
3. Set up the definite integral,
4. Integrate.
Ex. 1. Find the area in the first quadrant bounded by f ( x) 4 x x 2 and the x-axis.
Graph:
To find the boundaries, determine the x-intercepts: f ( x) 0 4 x x 2 0
x( 4 x) 0
x 0 or (4 x) 0 so x 0 and x 4
Therefore the boundaries are a 0 and b 4
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, b 4
Set up the integral: A f ( x)dx (4 x x 2 )dx
a 0
Solve:
4 4
1 2 1 3 2 1 3 1 3 1 3
4
0 (4 x x )dx 4 2 x 3 x 0 2 x 3 x 0 2 4 3 4 2 0 3 0
2 2 2
1 32
2 16 64 0
3 3
32
The area in the first quadrant under the curve f ( x) 4 x x is equal to
2
square units
3
Ex. 2. Find the area bounded by the following curves: y x 2 4, y 0, x 4,
Graph:
A
a b
Finding the boundaries:
y x 2 4, and y 0 implies x 2 4 0 so x 2x 2 0
x 2 or x 2
From the graph we see that x 2 is our boundary at a. The value x 2 is a solution to the
equation above but it is not bounding the area. (Here’s why the graph is an important tool to help
us determine correct results. Don’t skip this step!)
The other boundary value is given by the equation of the vertical line x 4,
Boundaries are: a 2, and b 4,
Set up the integral:
b 4
A f ( x)dx ( x 2 4)dx
a 2
64 8 64 8 56 32
16 8 16 8 8
3 3 3 3 3 3
32
The area bounded by the curves y x 4, y 0, x 4, is equal to
2
square units.
3
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