Here is a graph of the function y =r(t) =tan(cos(πt)+0.5)+2:
ibb.co/9qQjRvJ (Copy paste the link into browser, or type)
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 =
Jiskha does not allow cut and paste, just because your computer
allows it, (mine does too), on this page it does not work.
I also notice you might have meant:
ibb.com/9qQjRvJ
(I put in the extra m in .com), I ended up at some German website saying:
"Schade - diese Seite existiert leider nicht mehr"
Sorry, this page does not no longer exist.
I was able to graph your equation by entering it into
Desmos.com/calculator and from [0,12] you have 6 repeats of the same loop.
Obviously because of symmetry we could just use [0,1] then multiply
that answer by 12.
Do you want the area between the curve and the x-axis?
Looks like you just want us to do your work for you.
The url worked fine for me.
Since the interval [0,12] contains six symmetric copies of the curve, and you want 36 intervals, then the total area is 12 times the area under the curve in [0,1.5]
So all you need to do is use right-hand sums (why not left sums?) to get
A = 12 * 0.5 * (f(0.5)+f(1.0)+f(1.5))
To estimate the total area under the curve using a Riemann sum with 36 equal subdivisions and circumscribed rectangles, we can follow these steps:
1. Divide the interval [0, 12] into 36 equal subdivisions. Since the problem suggests using symmetry to make it easier, we can actually divide the interval [0, 6], since the function is symmetric about the y-axis.
2. Calculate the width of each subdivision. Since we have 36 subdivisions, the width of each subdivision will be (6/36) = 0.1667.
3. Choose a representative point for each subdivision. Since we are using circumscribed rectangles, we will choose the right endpoint of each subdivision as our representative point.
4. Evaluate the function at each representative point. In this case, the function is y = tan(cos(πt) + 0.5) + 2. So, for each subdivision, we need to calculate the value of the function at the right endpoint.
5. Calculate the area of each circumscribed rectangle. The area of each rectangle will be equal to the width of the subdivision multiplied by the value of the function at the right endpoint.
6. Sum up the areas of all the rectangles to find the total estimated area under the curve.
Note: Since I cannot see the image provided in the link, I cannot provide the exact numerical value for the area. However, you can follow these steps to calculate the estimated area using the given information.
To estimate the total area under the curve y = r(t) = tan(cos(πt) + 0.5) + 2 on the interval [0, 12] using a Riemann sum with 36 equal subdivisions and circumscribed rectangles, follow these steps:
1. First, observe that the hint suggests using symmetry to simplify the problem. Since the function has the tangent function as its main component, and the tangent function is an odd function, we can take advantage of its symmetry around the origin. This means we only need to consider the area in the positive x-axis and then double it.
2. Calculate the width of each subdivision, denoted by Δx. Since the interval [0, 12] is divided into 36 equal subdivisions, each subdivision has a width of (12 - 0) / 36 = 1/3.
3. Determine the x-values for the endpoints of each subdivision. The x-values for the endpoints can be calculated by starting from the left endpoint (x = 0) and adding Δx for each subsequent subdivision. In this case, the x-values for the 36 subdivisions are 0, 1/3, 2/3, 1, 4/3, 5/3, 2, 7/3, 8/3, 3, ...
4. Evaluate the function r(t) at each x-value to find the corresponding y-values. Use the given function y = r(t) = tan(cos(πt) + 0.5) + 2 to determine the y-value for each x-value obtained in step 3.
5. Find the maximum y-value among all the calculated y-values for each subdivision. Since we are using circumscribed rectangles, we want to find the function's maximum value within each subdivision.
6. Calculate the area of each circumscribed rectangle by multiplying the width (Δx) by the corresponding maximum y-value for each subdivision.
7. Sum up the areas of all the circumscribed rectangles to get an estimate of the total area under the curve.
Important Note: To get an accurate estimate, you may want to use a computer program or a spreadsheet that can handle the trigonometric calculations and summation easily.
Once you have completed these steps, you will obtain an approximation for the total area under the curve on the interval [0, 12]. This approximation can be provided by evaluating the Riemann sum using 36 subdivisions and doubling the result, as suggested by the hint.