Week 4 Lecture Notes, Lec01
Applications of Integrals
6.1. Area between curves
In this lecture we will review some applications of integrals. First, we start with the
area between curves.
Consider
f(x)
and
g(x)
are two functions of x. To find the area between
f(x)
and
g(x)
, from
a
to
b
, we can divide the areas into
n
smaller intervals with equal width
Δ
x
; then we approximate the area of each of the strips with a rectangle with base
Δ
x
and height
f(x)-g(x)
.
Then, area between the two curves can be obtained by adding the area of these
rectangles. When
n
approaches infinity, saying the rectangles are so thin that their
edges exactly fall ON the curves, then the exact area would be obtained.
∆? =
? − ?
𝑛
𝐴 = lim
?→∞
∑[?(?
𝑖
) − ?(?
𝑖
)]∆?
?
𝑖
= ∫ [?(?) − ?(?)]??
?
?
Example.
Find the area of the region bounded above by
? = ?
𝑥
, bounded below by
? = ?
,
and bounded by the sides by x = 0 and x = 1.
Solution
. In this kind of examples where details of the area between the curves as
well as the a and b bounds are given, all we have to do is to identify which curve
stands above the other one, then integrate.