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

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:

Step n: Using an n-sided polygon 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:

Step n: Using an n-sided polygon 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?

Area under the curve

Def: The area under a curve bounded by f(x) and the x-axis and the lines x = a and x = b is given by Whereand n is the number of sub-intervals

Conclusion: Inscribed rectangle Circumscribed rectangle 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.

3. Values for

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