expressions for upper and lower sums?

Question

i need to know how to write down the expressions for the upper and lower sums, Un and Ln, which would be used to calculate the area under the graph y = sin(x) over the interval [0, pi/2]

please just help me find the EXPRESSION needed to work it out, i dont actually have to evaluate these sums.

thanks heaps!

Answer 1

It's not clear to me what level you are at in this question. if you are taking calc I, you are probably using equal length intervals in your partition. Otherwise, probably not.

Well, sin x is an increasing function on that interval. So on each interval [x_i. x_{i+1}], the maximum value of sin x will be sin(x_{i+1}) and the minumum value will be sin(x_i). Each of these gets multiplied by the length of the interval (x_{i+1} -x_i) and summed.

In general, for a function f(x), let M_i be the maximum value of f on the interval [x_i,x_{i+1}] and m_i the minimum value. Then the upper and lower sums are given by

U(f,P)=sum M_i (x_{i+1}-x_i)

L(f,P)=sum m_i (x_{i+1}-x_i).

Answer 2

divide the interval in equidistant N+1 points x_0, x_1,...,x_N, distance between x_i, x_i+1 = d

define the rectangle L(i) as
d*sin(x_i), for i = 0, 1,...N-1

define the rectangle U(i) as
d*sin(x_i), for i = 1, 1,...N

L(i) is the rectange with upperleftcorner on x_i ( entire rectangle is under the graph of sin

U(i) is the rectangle with rightuppercorner on x_i

Sum L(i) <= Area <= Sum U(i)

take limit N --> inf gives the Area

Answer 3

Hi,
pl. contact me for solution.
I have master degree in mathematics and I have got thirteen ten years of teaching experience in Math at college level and currently I am working as a lecturer for an army engineering college author of math guide,
thankyou