Using an integral problem from the 2012 BC Calculus exam to practice with numerical integration

We had a snow day today and I was able to use some extra time with my son to talk about numerical integration techniques. I’d guess that I hadn’t seen the topic in at least 25 years, so it was fun to teach and review at the same time!

This evening I had my son review the ideas using an integration problem from the 2012 BC Calculus exam. I started by having him evaluate the integral exactly with no numerical integration techniques:

Next we used the trapezoidal rule and compared the answer to the exact answer –

The one thing I learned today was that the midpoint rule was roughly twice as accurate as the trapezoidal rule. Here my son used the midpoint rule and we do find that the answer is closer to the true value than the answer we found in the last video:

Finally, we used Simpson’s rule. Despite one little mistake in the middle, we ended up finding an answer that was surprisingly close to the exact asnwer.