# Shady areas; math portfolio type 1

This investigation is carried out in order to find a rule to approximate the area under a curve using trapeziums (trapezoids).

From calculus it is known that by the help of integration, the definite area under a curve could be calculated. In this case a geometrical method will be used in order to approximate the area under a curve.

The function g(x) = x2+3 is considered. The graph of g is shown below. The area under the curve from x = 0 to x = 1 is approximated by the sum of the area of two trapeziums.

In order to approximate the area under this curve, the composite trapezoidal rule is used. The composite trapezoidal rule is a numerical approach for approximating a definite integral.

g(x) dx = Area

x2 + 3 dx = Area

According to composite trapezoidal rule:

Area = ½ (width of strip) [first height + 2(sum of all middle heights) + last height]

Since there are two trapeziums from 0 to 1 in fig 1.1 hence the width of each strip: 1/2 = 0.5

So: Area = ½ (0.5) [3 + 2( 3.25) + 4]

Area = 3.37 area units (a.u.)

Increasing the number of trapeziums to five:

Approximating the area in this case with five trapeziums:

Width of every strip: 1/5 = 0.2

Area = ½ (1/5) [3 + 2(3.04 + 3.16 + 3.36 + 3.64) + 4]

Area = 3.34 a.u.

Finding the approximation for area under the same curve with increasing number of trapeziums:

1. 8 trapeziums:

Width of every strip: 1/8 = 0.125

Area = ½ (0.125) [3 + 2(23.1875) + 4]

= 3.3359375 a.u.

1. 12 trapeziums

Width of every strip: 1/12

Area = ½ (1/12) [3 + 2(36.51388889) + 4]

Area = 3.334490741 a.u

1. 16 trapeziums

Width of every strip: 1/16 = 0.0625

Area = ½ (1/16) [3 + 2(49.84375) + 4]

Area = 3.333984375 a.u.

Comparing the results from all the figures:

It is clearly observed that as the number of trapeziums increase, the uncertainty in approximating the area decreases since increasing number of segments of area under the curve are being taken into account. Thereof the value gets closer and closer to the actual value of the integral which is: 3.333333333 (from integration).

In order to find a general expression for the area under the curve of g(x) = x2 + 3 from x = 0 to x = 1, the following ...