Determine an upper/lower estimate for the area under the curve f(x) = cos x between x = 0 and x = pi/2 . Show how you arrive at this estimate.
This is the question that my teacher wanted us to answer and I'm not sure how to even start the process.
Should I use Reimann's Sum? If so how do I do it?
n
∑(pi/n) (cos(ipi/2n)
Also, wondering if this is a start?
t=1
n
∑(pi/n)(cos(pi/n)
t=1
corrected
It's a good start. Note that since you are using the right-side sum, and cos(x) is concave down and rising on that interval, the sum will be an upper estimate.
Use the left-side sum for a lower estimate.
Draw a few rectangles and the curve to see why this is true.
n
∑(pi/2n)(cos(t*pi/2n)
t=1
Okay I'm just confused about how many rectangles to use? Also, if I should only worry about [0,pi/2]
Use as many as you want. That's the n.
You can estimate the area with a single rectangle, of width pi/2 and height 1, but it won't be a very good estimate.
Also, my bad. I was thinking of sin(x) when I mentioned that the estimate was over- or under-. The reverse for cos(x), since it is decreasing on the interval.
And yes, you need to worry about the interval, since that was what was specified in the problem!!!
To determine an upper/lower estimate for the area under the curve f(x) = cos x between x = 0 and x = π/2, you can indeed use Riemann's Sum. Here's how you can approach it:
1. Start by dividing the interval [0, π/2] into smaller subintervals. Let's denote the number of subintervals as n. Each subinterval will have a width of Δx = (π/2 - 0)/n = π/(2n).
2. Pick a representative value within each subinterval. You can use the left endpoint, right endpoint, or midpoint of each subinterval. Since it's not specified, let's use the left endpoints in this explanation.
3. Write the Riemann sum for the area approximation:
A = ∑[i=1 to n] f(xi)Δx
In this case, f(xi) is cos(xi) and Δx is π/(2n). So the equation becomes:
A = ∑[i=1 to n] cos(xi) * (π/(2n))
4. Substitute xi:
Let's build a table of xi values for each subinterval:
xi = (i - 1) * Δx
For example, when n = 4:
x1 = (1 - 1) * π/(2*4) = 0
x2 = (2 - 1) * π/(2*4) = π/(2*4)
x3 = (3 - 1) * π/(2*4) = π/(2*2*4)
x4 = (4 - 1) * π/(2*4) = 3*π/(2*4)
Repeat this process for higher values of n.
5. Calculate the sum:
Substitute the xi values in the Riemann sum equation:
A = cos(x1) * (π/(2n)) + cos(x2) * (π/(2n)) + cos(x3) * (π/(2n)) + ... + cos(xn) * (π/(2n))
For each value of n, calculate the sum.
6. Determine upper/lower estimates:
Since you are asked to find an upper and lower estimate, you will need to find two Riemann sums: one using the maximum and one using the minimum value of cos x in each subinterval. The maximum value of cos x is 1, and the minimum value is -1.
For the upper estimate, replace each cos xi with 1, and for the lower estimate, replace each cos xi with -1. Then calculate the sums as described in step 5.
The upper estimate will give you an upper bound on the area under the curve, and the lower estimate will give you a lower bound on the area.
Following these steps should help you arrive at the upper and lower estimates for the area under the given curve.