Section 6.6 Area Between Two Curves
Area Between Two Curves:
The area A of the region bounded by the curves y = f (x) and
y = g(x) and the lines x = a and x = b, where f and g are continuous and f (x) ≥ g(x) for all x in [a, b]
is given by the definite integral
Z
b
[f (x) − g(x)] dx
A=
a
Area of a Curve under the x-Axis:
If the graph of y = f (x) is below the x-axis on [a, b],
then the area A below the x-axis and above the graph of y = f (x) on [a, b] is
Z
A=−
b
f (x) dx
a
1. Determine the area that is bounded by the following curve and the x-axis on the interval below.
(Round answer to three decimal places.)
y = x2 − 9,
−6 ≤ x ≤ 2
2. Determine the area that is bounded by the following curve and the x-axis on the interval below.
(Round answer to three decimal places.)
y = e−3x ,
−2 ≤ x ≤ 1
3. Determine the area that is bounded by the graphs of the following equations.
y = x3
y = 64x,
4. Determine the area that is bounded by the graphs of the following equations. (Round answer to
three decimal places.)
y = 3x,
y = 9x − x2
5. Determine the area that is bounded by the graphs of the following equations on the interval below.
(Round answer to three decimal places.)
y = x2 + 7x,
y = 8x + 56,
−2 ≤ x ≤ 2
2
Fall 2016,
©
Maya Johnson
6. Determine the area that is bounded by the graphs of the following equations. (Round answer to
three decimal places.)
y = −x2 ,
y = x3 − 6x
7. The graph of f is shown. Evaluate each integral by interpreting it in terms of areas.
Z
28
f (x) dx
(a)
20
Z
36
f (x) dx
(b)
0
3
Fall 2016,
©
Maya Johnson