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) = 2.01745506t = 2/3, r(t) = 2t = 1, r(t) = 1.45369751These 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) = 1.823717523I can multiply the area by 12 to find the area under the entire function:1.823717523 * 12 = 21.88461028This is not an answer choice, so I clearly did something wrong! Can someone help me find my mistake?

Accepted Solution

"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.