In the Integrals with Riemann Sums article, we used rectangles to estimate integrals.
Another common shape used to estimate integrals visually is trapezoids!
This operates very similarly to Riemann Sums, as we can see by the formula for Trapezoid Rule! We're just summing the area of each trapezoid...
...which can be simplified to this...
...and visualized like so:
Calculating the base
From the Integrals with Riemann Sums article, we utilized this equation...
...to calculate the base of each rectangle to be equal to 5.
The same calculation applies here, as we're covering between the upper bound (b) of 20 and lower bound (a) of 0 with 4 (n) shapes... in this case trapezoids!
Therefore, we can update our Trapezoid Rule equation like so:
Calculating the heights
To find the height of these rectangles, we need to determine the value of f(x) at each of the trapezoidal corners that touch the function.
We can represent this mathematically in our Trapezoid Rule equation like so:
When we calculate these height values, we get the following...
...which can be plugged into our Trapezoid Rule equation like so:
When we solve, we get the following final estimation for the integral of f(x) between 0 and 20 of 168.75!
Which is more accurate?
When compared to the 3 Riemann Sums methods...
| Method | Solution |
|---|---|
| Left Riemann Sums | 143.75 |
| Right Riemann Sums | 193.75 |
| Midpoint Riemann Sums | 165.625 |
| Trapezoid Rule | 168.75 |
| Exact Answer | 166.667 |
...we find that the Trapezoid Rule, like the Midpoint Riemann Sums, is quite accurate to the exact answer.
This is typical!
The Midpoint Riemann Sums and Trapezoid Rule are typically the most accurate ways to estimate an integral of a function!