Section 4-2

Area and Integration

Basic Geometric Figures

rectangle

triangle

parallelogram

Area of a Circle: Archimedes (212 BC)

Approximating the area of a circle with polygons

We notice that the diagonals of the square are the same as the diameter of the circle, and so have length

Thus the sides of the square, AB, BC which are equal of course, must have length 2r

(Since ABC is a right triangle, whose hypotenuse is AC). Thus the area of the inscribed square is

r2

If we use a polygon with more sides to try to approximate the area of the circle, we would hope to get a better result. So consider an approximation which uses the hexagon, as shown in the diagram.

To find the area of the hexagon, we might subdivide it into six triangles, whose area is easily computed if we know the height and the base of any one of the (all equal) triangles. One such triangle is shown in the diagram at right:

Suppose we increase the number of sides in the polygon and look at the general case, in which there are n sides. Then the area of the n sided polygon will be n times the area of one of the triangles, i.e.

                                                                       

where b and h are, respectively the base and height of one of the triangles shown in this picture. Now note what happens as the number of sides, n increases:

                                                         

             

              

Suppose we increase the number of sides in the polygon and look at the general case, in which there are n sides. Then the area of the n sided polygon will be n times the area of one of the triangles What happens as the number of sides, n increases?

                                                         

         

                   The perimeter of the polygon, which, as n increases, becomes

closer and closer to the circumference of the circle. Further, the height of the triangle, h   approaches the radius of the circle, so that, as we approximate the circle by a polygon with more sides, i.e. a greater number of (thinner) triangles, we find that the area approaches:

                                                                        

Exhaustion Method: is a limiting process in which the area is squeezed between an n-sided inscribed polygon and an n-sided circumscribed polygon as n increases, the area of the polygon becomes a better approximation of the area of the circle.

Area Under a Curve

Add the area of the rectangles to approximate the area under the curve

Each rectangle has a height f(x) and a width dx

1. How can we get

a closer approximation?

http://www.ugrad.math.ubc.ca/coursedoc/math101/notes/integration/area.html

Area under the curve

Def: The area under a curve bounded by f(x) and the x-axis and the linesx = a and x = b is given by

Where and n is the

number of sub-intervals

n

i

dxxfn 1

)(lim

n

abdx

Conclusion:

n

i

n

i

dxxfregionofareadxxf1

21

1)(

Inscribed rectangle

Circumscribed rectangle

http://archives.math.utk.edu/visual.calculus/4/areas.2/index.html

The sum of the area of the inscribed rectangles is called a lower sum, and the sum of the area of the circumscribed rectangles is called an upper sum

2.

6

1

2 1n

n

3.

s(n)81

n4n2 n 1 2

4

s(n)81

n4i3

Values for

260, 32 pgonfoundareii

Assignment:

Page 267 # 2,3,17,38,39 and 40