An Overview of the Area Problem

n = number of intervals arbitrary

Δx = size of intervals

i = which interval 1 to n

Finding the area could be broken down into the following parts:

Here, x1 = 0, n = 4, and Δx = 0.5

Lower estimate → left endpoints approximation method - LAM

Upper estimate → right endpoints approximation method - RAM

Here, x1 = 0, n = 4, and Δx = 0.5

Average estimate → mid endpoints approximation method -

MAM

Here, x1 = 0, n = 4, and Δx = 0.5

At some point we will use these formulas (pg.375 and pg. 376)

At some point you may want to use this formula:

How would we find the actual area?

Example Find the area of the region under the curve using: LAM

y = x2 [0, 2] n = 4

0 0.5 1 1.5 2

Example Find the area of the region under the curve using: RAM

y = x2 [0, 2] n = 4

0 0.5 1 1.5 2

Example Find the area of the region under the curve using: MAM

y = x2 [0, 2] n = 4

0.25 0.75 1.25 1.75

The Antiderivative method for Finding Areas

a x

A (x)

A’ (x) = f(x)

Let’s see how that worksExample: 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.

-1 x

y =2

2

x + 1

A = L* w

A = 2 * (x + 1)

A = 2 x + 2

A’ = 2 = f (x)

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.

A =1/2 * b* h

A =1/2 * (x + 1) * (x + 1)

A =1/ 2 (x2 + 2x+ 1)

A = x2/2 + x + 1/2

y = x + 1

x + 1

-1 x x + 1

A’ = x + 1 = f (x)

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.

A =1/2 *( b1 + b2) * h

A =1/2 *[ (2x + 3) + (2*(-1) + 3) ]* (x + 1)

A =1/2 * (2x + 4) * (x + 1)

A = 1/2 * (2x2 + 6x + 4)

A’ =2x + 3 = f (x)

y = 2x + 3

2x + 3

-1 x

x + 1

A = x2 + 3x + 2