Section 6.2: Area
Suppose that f is a continuous function on the interval [a, b]. What is the area of the
region S that lies under the curve y = f (x) from x = a to x = b?
To approximate the area of this region, we begin by subdividing the interval [a, b] into n
subintervals by choosing partition points x0 , x1 , x2 , . . . , xn such that
a = x0 < x1 < x2 < · · · < xn−1 < xn = b.
Then the n subintervals are
[x0 , x1 ], [x1 , x2 ], . . . , [xn−1 , xn ].
This subdivision is called a partition of [a, b], and denoted by P . The length of the ith
subinterval [xi−1 , xi ] is
∆xi = xi − xi−1 .
The length of the longest subinterval is called the norm of P and denoted by ||P ||. That is,
||P || = max{∆x1 , ∆x2 , . . . , ∆xn }.
Then we choose a representative point x∗i in each subinterval [xi−1 , xi ] and construct an
approximating rectangle Ri with base ∆xi and height f (x∗i ).
The area of the region S is approximated by the Riemann sum
A≈
n
X
f (x∗i )∆xi .
i=1
1
Example: Estimate the area under the graph of f (x) = 16 − x2 on [0, 4] using four approximating rectangles and left endpoints. Is your estimate an underestimate or overestimate?
2
Example: Estimate the area under the graph of f (x) = 4 cos x on [0, π/2] using four approximating rectangles and right endpoints. Is your estimate an underestimate or overestimate?
3
Example: Estimate the area under the graph of f (x) = ex on [−2, 2] using four approximating
rectangles and midpoints.
4
Theorem: If f (x) ≥ 0 on [a, b], then the true area under the graph of y = f (x) on [a, b] is
A = lim
||P ||→0
n
X
f (x∗i )∆xi
= lim
n→∞
i=1
n
X
i=1
f (x∗i )
b−a
n
.
Example: Find the area under the curve y = x2 + 3x − 2 on [1, 4] using a Riemann sum with
equal subintervals and right endpoints.
5
Example: Determine a region whose area is equal to the given limit.
r
n
X
3
3i
(a) lim
1+
n→∞
n
n
i=1
n
X
π
πi
(b) lim
sin
n→∞
4n
4n
i=1
6