Area Question!

Question

A left sum ad a right sum are used to estimate the area between the graph of the function f(x)=sinx and the x axis from x=0 to x=1.5 Which of the two sum gives an underestimate of the value of the area? Justify your answer and show your work

Answer 1

pi > 3

pi/2 > 1.5

 

Since sin(x) is an increasing function on [0,pi/2], the inequality above also shows that it is an increasing function on [0,1.5].  On this interval, for any two x-values a and b, if b>a, then sin(b) > sin(a). 

 

This means that rectangles whose height is given by the left endpoint of an interval will be entirely underneath sin(x), and rectangles whose height is given by the right endpoint will be entirely above sin(x)...except at the endpoints themselves, which doesn't affect area over/undershoot.

 

The area produced by rectangles whose height is less than or equal to sin(x) on any given interval is also less than or equal to the area under sin(x) on the same interval.  Replace "less than or equal to" with "greater than or equal to", and you have another truthful statement.

 

From this, it can be seen that rectangles whose height is given by left endpoints of intervals will undershoot the actual area under sin(x).

 

That is the justification; as for the work that must be shown, demonstrate (either from a simple sketch or by use of the derivative) that sin(x) is an increasing function on [0,1.5].

 

Hope this helps, and happy holidays!

 

 

 

 

Answer 2

The area of an integral is estimated by calculating the mean of the upper and lower sums (i.e. the total area of the circumscribed and inscribed area rectangles) within left and right bounds, thus the lower sum (i.e. the inscribed area rectangles) would underestimate the area while the upper sum (i.e. the circumscribed area rectangles) would overestimate the area.

Given the area of ∫₀^1.5sin(x) = [-cos(x)]₀^1.5 = -cos(1.5)+cos(0) = .929263, the lower sum would be any value less than this dependant upon the number of area rectangles used.