1. An Overview of the Area
Problem

2. Finding the area could
be broken down into the
following parts:
n = number of intervals
Δx = size of intervals
i = which interval
arbitrary
x2 − x1
n
1 to n

3. Lower estimate → left endpoints
approximation method - LAM
Here,
x1 = 0,
n = 4, and
Δx = 0.5
n
∑ f (x)• ∆x
i=1

4. Upper estimate → right endpoints
approximation method - RAM
Here,
x1 = 0,
n = 4, and
Δx = 0.5
n
∑ f (x)• ∆x
i=1

5. Average estimate → mid
endpoints approximation method MAM
Here,
x1 = 0,
n = 4, and
Δx = 0.5
n
∑ f (x)• ∆x
i=1

6. At some point we will use these formulas (pg.375 and pg. 376)
At some point you may want to use this formula:

7. How would we find the actual area?
Area = lim An
n→∞

8. Example Find the area of the
region under the curve using: LAM
y = x2
∆x =
[0, 2]
n=4
2−0
= 0.5
4
A = h ×l = f (x)×∆x
A1 = f (0)×0.5 = 0
A2 = f (0.5)×0.5 = 0.125
A3 = f (1)×0.5 = 0.5
A4 = f (1.5)×0.5 = 1.125
Atotal = A1 + A2 + A3 + A4
0
0.5
1
1.5
2
Atotal = 0 + 0.125 + 0.5 +1.125 = 1.75

9. Example Find the area of the
region under the curve using:
RAM
y = x2 2 − 0[0, 2]
∆x =
4
= 0.5
n=4
A = h ×l = f (x)×∆x
A1 = f (0.5)×0.5 = 0.125
A2 = f (1)×0.5 = 0.5
A3 = f (1.5)×0.5 = 1.125
A4 = f (2)×0.5 = 2
Atotal = A1 + A2 + A3 + A4
0
0.5
1
1.5
2
Atotal = 0.125 + 0.5 +1.125 + 2 = 3.75

10. Example Find the area of the
region under the curve using: MAM
y = x2
∆x =
[0, 2]
n=4
2−0
= 0.5
4
A = h ×l = f (x)×∆x
A1 = f (0.25)×0.5 = 0.0325
A2 = f (0.75)×0.5 = 0.28125
A3 = f (1.25)×0.5 = 0.78125
A4 = f (1.75)×0.5 = 1.53125
Atotal = A1 + A2 + A3 + A4
0.25
0.75
1.25
1.75
Atotal = 0.0325 + 0.28125 + 0.78125 +1.53125 = 2.62625

11. The Antiderivative method for
Finding Areas
A’ (x) = f(x)
A (x)
a
x

12. Let’s see how that works
Example: Find the area A(x) between the
graph of f(x) = 2 and the interval [a, x] = [-1
,x], and find the derivative A’(x) of this area
function.
y =2
A = L* w
A = 2 * (x + 1)
2
A=2x+2
-1
x
x+1
A’ = 2 = f (x)

13. More Examples !!!
Example: Find the area A(x) between the
graph of f(x) = x + 1 and the interval [a, x] = [1 ,x], and find the derivative A’(x) of this area
function.
y=x+1
A =1/2 * b* h
x+1
A =1/2 * (x + 1) * (x + 1)
A =1/ 2 (x2 + 2x+ 1)
-1
x
x+1
A = x2/2 + x + 1/2
A’ = x + 1 = f (x)

14. One More Example !!!
Example: Find the area A(x) between the
graph of f(x) = 2x + 3 and the interval [a, x] =
[-1 ,x], and find the derivative A’(x) of this area
function.
y = 2x + 3
A =1/2 *( b1 + b2) * h
2x + 3
A =1/2 *[ (2x + 3) + (2*(-1) + 3) ]* (x + 1)
A =1/2 * (2x + 4) * (x + 1)
A = 1/2 * (2x2 + 6x + 4)
-1
x
x+1
A = x2 + 3x + 2
A’ =2x + 3 = f (x)