Math 152 Class Notes
September 8, 2015
7.1 Area Between Curves
ˆ
Recall that when f (x) ≥ 0 on an interval [a, b], then
the graph of f from x = a to x = b.
b
f (x)dx gives the area under
a
π
4
Example 1. Find the area bounded by y = sin x, y = 0, x = 0 and x = .
The area between the curves y = f (x), y = g(x) and the lines x = a and x = b, where
f (x) ≥ g(x) for all x in the interval [a, b] is
ˆ
b
(f (x) − g(x))dx
A=
a
It can be remembered as
ˆ
Area =
b
[ top − bottom ] dx
a
1
x
Example 2. Find the area bounded by y = , y =
1
, x = 1 and x = 2.
x2
Example 3. Find the area bounded by y = x2 and y =
2
.
x2 + 1
Example 4. Find the area bounded by y = |x| and y = x2 − 2.
If we are asked to nd the area bounded by the curves y = f (x), y = g(x) where
f (x) ≥ g(x) for some values of x but g(x) ≥ f (x) for other values of x, we must split
the integral at each intersection point.
Example 5. Find the area bounded by y = cos x, y = 0, x = 0 and x =
2π
.
3
Example 6. Find the area bounded by y = sin x, y = cos x, x = −
π
π
and x = .
2
2
Example 7. Find the area bounded by y = ex , y = e−x , x = −2 and x = 1.
The area between the curves x = f (y), x = g(y) and the lines y = c and y = d, where
f (y) ≥ g(y) for all y in the interval [c, d] is
ˆ
d
(f (y) − g(y))dy
A=
c
It can be remembered as
ˆ
Area =
d
[ right − left ] dy
c
1
x
Example 8. Find the area bounded by y = , x = 0, y = 1 and y = 2.
Example 9. Find the area bounded by y =
√
x, y = x2 , x = 0 and x = 1.
Example 10. Find the area bounded by y 2 = x and x − 2y = 3.