Here is a graph of the function y = r(t) = tan(cos(πt) + .5) + 2:
ibb.co/9qQjRvJ (Link to image)
Estimate the total area under this curve on the interval [0, 12] with a Riemann sum using 36 equal subdivisions and circumscribed rectangles. Hint: Use symmetry to make this problem easier. You get Area =
Can not see it. Copy and paste does not work on this site.
Copy paste seems to work fine on computer.
If not, you could try typing, though it would be a pain.
I appreciate the response regardless.
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To estimate the total area under the curve using a Riemann sum, we can divide the interval [0, 12] into equal subdivisions and approximate the area of each subdivision using circumscribed rectangles.
Since the given function has symmetry, we can take advantage of this to simplify the calculations. The interval [0, 12] can be divided into 36 subdivisions, which means each subdivision will have a width of (12-0)/36 = 1/3.
Next, we need to calculate the height of each circumscribed rectangle. For each subdivision, we will calculate the maximum value of the function within that interval and use it as the height of the rectangle.
To find the maximum value in each subdivision, we can evaluate the function at the endpoints and the midpoint, and then select the maximum value among these three points.
Let's walk through the process step by step:
1. Divide the interval [0, 12] into 36 equal subdivisions, each with a width of 1/3. Mark the endpoints of each subdivision.
2. Evaluate the function at the endpoints and the midpoint of each subdivision.
For example, to find the maximum value in the first subdivision [0, 1/3]:
- Evaluate the function at t = 0, t = 1/3, and t = 1/6.
- Take the maximum value among these three evaluations as the height of the rectangle in the first subdivision.
3. Repeat step 2 for each of the remaining subdivisions, finding the height of the rectangle by comparing the function values at the endpoints and midpoint of each subdivision.
4. Once you have determined the height of each rectangle for all 36 subdivisions, calculate the area of each rectangle by multiplying the width (1/3) by the corresponding height.
5. Finally, sum up all 36 rectangle areas to estimate the total area under the curve on the interval [0, 12].
Note: Remember that the Riemann sum is an approximation, and the more subdivisions you use, the closer your estimate will be to the actual area.