Understanding the trapezoidal rule
Walk through an example using the trapezoid rule, then try a couple of practice problems on your own.
By now you know that we can use Riemann sums to approximate the area under a function. Riemann sums use rectangles, which make for some pretty sloppy approximations. But what if we used trapezoids to approximate the area under a function instead?
Key idea: By using trapezoids (aka the "trapezoid rule") we can get more accurate approximations than by using rectangles (aka "Riemann sums").
An example of the trapezoid rule
Let's check it out by using three trapezoids to approximate the area under the function on the interval .
Here's how that looks in a diagram when we call the first trapezoid , the second trapezoid , and the third trapezoid :
Recall that the area of a trapezoid is where is the height and and are the bases.
Finding the area of
We need to think about the trapezoid as if it's lying sideways.
The height is the at the bottom of that spans to .
The first base is the value of at , which is .
The second base is the value of at , which is .
Here's how all of this looks visually:
Let's put this all together to find the area of :
Simplify:
Finding the area of
Let's find the height and both of the bases:
Plug in and simplify:
Find the area of
Finding the total area approximation
We find the total area by adding up the area of each of the three trapezoids:
Here's the final simplified answer:
You should pause here and walk through the algebra to make sure you understand how we got this!