Definite Intergrals

Question

Approximate the area of the region bounded by the graph of f(x)= cos (t/2-7pi/8) and the t-axis on [7pi/8, 15pi/8] with n=4 sub intervals. Use the midpoint of each sub interval to determine the height of each rectangle.

 

The approximate area of the region is _________. Round to two decimal places.

Answer 1

Ahh the good ole "approximation" problems:

 

This is much easier to do on paper with a nice pencil than on a chat board but let's see how this goes:

 

You have some domain restriction namely between [7pi/8, 15pi/8] which is a total domain of 15pi/8 - 7pi/8 = 8pi/8 or pi.

 

Now you need to break this into 4 intervals of pi/n = pi/4.

 

This tells you that the width of each rectangle will be pi/4. Now to find the height which will be given by function cos(t/2 - 7pi/8)

The first thing you need to do is draw the graph(a calculator can help) but it's pretty simple, it's a cos graph with a period of 4pi and shifted to right by 7pi/8.

 

Now we need to make our rectangles, each rectangle's height will be cos(x/2 - 7pi/8) where x is the midpoint of each interval. For example, for the first interval of 7pi/8 to 9pi/8, the midpoint is (7 pi/8 + 9pi/8)/2 or pi. This can be complicated by the fact that any area UNDER the x axis in negative and therefore needs to subtracted away but hopefully this explanation helped(I do not know if this particular graph does dip under the x axis on the range given). Just do this procedure 4 times, 1 for each interval, and sum up their areas to get the correct answer.

 

Look up Riemann Sum approximation of the area under a curve for a detailed explanation!