12.5b Area of a Plane Region

Let's let n = 4

so the width = (1 -

height =

[-1, -1/2] = 1.75

[-1/2, 0] = 2

[0, 1/2] = 1.75

[1/2, 1] = 1

so the Area when f(x) = 2 - x

(1/2)(1.75) + (1/2)(2) + (1/2)(1.75) + (1/2)(1) = 3.25

but to find the area with more rectangles, let's use the above formulas:

Area of a Region bounded by:

*x*=

*a*,

*x*=

*b*,

*y*= 0 and

*y*= f (

*x*) where f(

*x*) is less than or equal to 0.

The boundary of the area is from x

_{0}= a to x_{n}= b.What is the area under the curve?

**George Riemanns**- German who came up with technique to find the orange area.

1. Subdivide the interval from

*a*to*b*into smaller intervals. This is called a partition of [a,b] specifically:a = x

_{0}≤ x_{1}≤ x_{2}≤ x_{3}... ≤ x_{n - 1}≤ x_{n}= b2. Choose a point w

_{i}in interval [x_{i - 1}, x_{i}] , i = 1 ... nso...

when i = 1, w

_{1}is contained in [x_{0}, x_{1}]when i = 2, w

_{2}is contained in [x_{1}, x_{2}]3. Let Δ x

_{i}= x_{i}- x_{i-1}for i = 1 ... ni= 1; Δ x

_{1}= x_{1}- x_{0}i = 2; Δ x

_{2}= x_{2}- x_{1}Δ x

_{i}= length of the i^{th}subinterval.4. Form f(w

_{i})(Δx_{i}) for i = 1 ... n5. The area of the region:

A = f (w

_{1})(Δ x_{1}) + f(w_{2})(Δ x_{2}) + ... + f (w_{n})(Δx_{n})This is called a

**Riemann's Sum**.So by increasing the number of rectangles that you make, you can obtain a closer and closer approximation

**Therefore to find the Area of a Plane Region:**

Let "f" be continuous function and nonnegative on the interval [a, b]. The Area A of the region bounded by the graph of "f", the x-axis (y = 0), and the vertical lines x = a and x = b is

Area of a rectangle = height times width

Area of a rectangle = height times width

**Example 1:**

Find the Area of the Region: f(x) = 2 - x

^{2}, -1 £ x £ 1

Let's let n = 4

so the width = (1 -

^{-}1)/4 = 1/2

height =

[-1, -1/2] = 1.75

[-1/2, 0] = 2

[0, 1/2] = 1.75

[1/2, 1] = 1

so the Area when f(x) = 2 - x

^{2}and is divided up into 4 rectangles =

(1/2)(1.75) + (1/2)(2) + (1/2)(1.75) + (1/2)(1) = 3.25

but to find the area with more rectangles, let's use the above formulas:

Therefore to find the area, use the summation formulas:

To check this on the graphing calculator:

put in y

Now on your homescreen, math arrow down to #9 fnInt (Y

should give you 10/3.

In the calculator (function , variable, lower bound, upper bound) = area under the curve.

put in y

_{1}= 2 - x^{2}Now on your homescreen, math arrow down to #9 fnInt (Y

_{1}, x, -1, 1) entershould give you 10/3.

In the calculator (function , variable, lower bound, upper bound) = area under the curve.