Question

I have attached a picture of a question I am stuck on. It is about Riemann sums. I have attempted the question but did not get the correct answer :( Here is my work:
I see that this graph has the same area repeating 12 times.

I did (b - a)/n, or (12 - 0)/36, and found that the width of each rectangle should be 1/3. Because we are asked to find the area using "circumscribed angles," I will use right-endpoint rectangles on the first part of the graph that shows a decreasing interval from 0 to 1:

t = 1/3, r(t) = 2.01745506
t = 2/3, r(t) = 2
t = 1, r(t) = 1.45369751

These values for r(t) represent heights of the rectangles. I can add them and multiply by the width of each rectangle:

1/3(2.01745506 + 2 + 1.45369751) = 1/3(5.47115257) = 1.823717523

I can multiply the area by 12 to find the area under the entire function:

1.823717523 * 12 = 21.88461028

This is not an answer choice, so I clearly did something wrong! Can someone help me find my mistake?

Answer 1

"Circumscribed rectangles" means that any Riemann Sum (left or right) must overestimate the area under the curve. So, a Right-Riemann sum would underestimate the area under the curve, and that's where you made your mistake. You will use the Left-Riemann Sum to approximate the area under the curve r(t) = tan(cos(xt) + 0.5) + 2

Or, you could use u-substitution to get the exact area under the curve from [0, 12] - but I would do as the problem says. If you want me to that, DM me.