Blog 4 Ashah Alkhater

Hello everyone, I am professor “YYY” and today I am going to explain trapezoidal rule to calculate the area under a curve. Before going to calculate, I will explain the Riemann Sum Review first.

Riemann Sum Review

Let's consider the ramp shown in figure which is characterized by the equation y = x^2 + 1. The area under this curve resembles how much dirt can be remove if someone go from 0 to 2 with a mop. In order to find out how much dirt is going to be removed underneath this ramp, Riemann sum should be used. It considers only one slice along this curve. According to Riemann sum, at first the height somewhere along this curve is measured, and that height is multiplied by 2 meters. That's the distance in x. The height times the width here would the cross-sectional area and tell about how much dirt would be removed.

If a measurement has been made at the far left side, it is a left-side Riemann sum, it gives an area of 2 because the height on the left side is 1 multiplied by the width, which is 2. If a midpoint Riemann sum is considered, area would be estimated at 4. If a right-side Riemann sum is used, the cross-sectional area would be estimated to be 10. This sounds absolutely fantastic, but none of these projections look right.

Area of a trapezoid flipped on its side

So, rather than taking multiple slices and doing a Riemann sum with two different areas, it is not wise to use rectangles and instead estimate this area with a trapezoid.

Using Trapezoids to Estimate Area

The area of a trapezoid is equal to the height of the trapezoid times the average of the two parallel sides.

Area = (Height) * (w sub 1 + w sub 2) / 2

Where, w sub 1 is the length of short edge, and w sub 2 is the long side. Even flipping trapezoid sides still gives the same area, but the height is going along horizontally and w sub 1 on the left side of the trapezoid and w sub 2 on the right side of the trapezoid. Well, this is the function to calculate area.

Therefore, a trapezoid outline is going between the left side, 0, and the right side, 2.

In this case, w sub 1 is the height on the left side, w sub 2 is the height on the right side, and the height is actually the distance between 0 and 2 on the x-axis. If these points are plugged into the area formula, the area equals the value of the function on the left side plus the value of function on the right side all divided by 2 times my delta x. That's the difference between the left-side value of x and the right-side value of x. So in this case, it's 2 - 0.

Using a trapezoid to estimate area on the graph of the function

By plugging in the points for y = x^2 + 1 from 0 to 2. f(x) on the left side is equal to f(0). f(x) on the right side is f(2) - all of that divided by 2 times my delta x, which is 2 - 0. If 0 is plugged into the function y = x^2 + 1, results is 1. If 2 is plugged into that function, result is 5. So the area becomes ((1 + 5) / 2) * 2, and that's just 6.

It can be even better estimated by dividing this into two slices, and take the trapezoid area of two different slices and add them up to get the total sum.

In this way, the first area goes from f(0) to f(1), so the delta x is 1 - 0, and the second area goes from f(1) to f(2), so the delta x is 2 - 1. If the values are plugged in for f(0), f(1) and f(2), the area under the curve is estimated to be 9 / 2.

Similarity, the area can be estimated more accurately if number of the segments becomes higher. Theoretically, the infinite number of segments can give the actual area under the curve.