This document discusses calculating the area between curves. It defines the area between two curves f(x) and g(x) from x=a to x=b as the integral from a to b of [f(x) - g(x)]dx. When f and g alternate being above and below each other, the region is divided into parts and the areas added. It also discusses setting x as a function of y and calculating area using integrals with respect to y. Examples are provided to illustrate the techniques.

1. Chapter 6: Applications of Integration
Section 6.1: Areas Between Curves
Alea Wittig
SUNY Albany

2. Outline
Definition of Area Between Curves
Examples
Region S with Alternating Top and Bottom Curves
Example
Region S with Left and Right Curves
Examples

3. Definition of Area Between Curves

4. Region S Between Two Curves
▶ Let y = f (x) and y = g(x)
be two curves such that
f (x) ≥ g(x) between the
vertical lines x = a and
x = b, and f and g are
continuous on [a, b].
▶ Let S be the region
enclosed by these two
curves and A the area of S.

5. Computing the Area of S - Approximating with Rectangles
▶ Divide S into n strips of
equal width and
approximate the ith strip by
a rectangle with base △x
and height f (x∗
i ) − g(x∗
i ).
▶ Here x∗
i is any value in the
interval [xi−1, xi ].

6. Computing the Area of S - Approximating with Rectangles
→
▶ The sum of the areas of the n rectangles will approximate the
exact area A of the region S for large n.
▶ As we take n → ∞, the sum of the areas of the rectangles will
give the exact area A of S.

7. Computing the Area of R - Approximating with Rectangles
▶ The n rectangles have
equal width △x = b−a
n
height = f (x∗
i ) − g(x∗
i ).
▶ The area of a rectangle is
[height]·[width]=
[f (x∗
i ) − g(x∗
i )]△x
A ≈
n
X
i=1
[f (x∗
i ) − g(x∗
i )]△x for large values of n =⇒
A = lim
n→∞
n
X
i=1
[f (x∗
i ) − g(x∗
i )]△x =
Z b
a
[f (x) − g(x)]dx

8. Area Between Curves
▶ So if f (x) ≥ g(x) for all a ≤ x ≤ b, then the area of the
region between f and g on the interval a ≤ x ≤ b is
A =
Z b
a
[f (x) − g(x)]dx
▶ Denoting the top function by yTop or yT , and the bottom
function by yBottom or yB we have
A =
Z b
a
[yT − yB]dx

9. Examples

10. Example 1
Find the area of the region bounded above by y = ex , below by
y = x and on the sides by x = 0 and x = 1
Video Solution 1

11. Example 2
Find the area of the region enclosed by the parabolas y = x2 and
y = 2x − x2.
Video Solution 2

12. Region S with Alternating Top and Bottom
Curves

13. Suppose we are asked to find the area between two curves f and g
where f (x) ≥ g(x) for some values of x and g(x) ≥ f (x) for other
values of x.
Let Ai be the area of the region
Si for i = 1, 2, 3.
S = S1 ∪ S2 ∪ S3
A = A1 + A2 + A3
=
Z x1
a
[f (x) − g(x)]dx + . . .
Z x2
x1
[g(x) − f (x)]dx + . . .
Z b
x2
[f (x) − g(x)]dx

14. In general, since
|f (x) − g(x)| =
(
f (x) − g(x) if f (x) ≥ g(x)
g(x) − f (x) if g(x) ≥ f (x)
we have that the area between the curves y = f (x) and y = g(x)
and between x = a and x = b is
A =
Z b
a
|f (x) − g(x)|dx

15. Example

16. Example 3
Find the area bounded by the
curves y = sin x and y = cos x,
x = 0, and x = π/2.
Video Solution 3

17. Region S with Left and Right Curves

18. Some regions are best treated by regarding x as a function of y.
If we have a region bounded by
the following,
x = f (y)
x = g(y)
y = c
y = d
where f and g are continuous
and f (y) ≥ g(y) for c ≤ y ≤ d
then its area is
A =
Z d
c
[f (y) − g(y)]dy

19. Examples

20. Example 4
Find the area enclosed by the curves x = 9 − y2 and x = y2 − 9
Video Solution 4

21. Example 5
Find the area of the region (in
quadrant 1) enclosed by the
curves y = 1/x, y = x, and
y = 1
4x using
a. x as the variable of
integration and
b. y as the variable of
integration.
Video Solution 5