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.

Question

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

Answer 1

n
∑(pi/n)(cos(pi/n)
t=1

corrected

Answer 2

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.

Answer 3

n
∑(pi/2n)(cos(t*pi/2n)
t=1

Answer 4

Okay I'm just confused about how many rectangles to use? Also, if I should only worry about [0,pi/2]

Answer 5

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!!!