Calculus Question

Question

The table gives the values of a function obtained from an experiment. Use them to estimate 9	f(x) dx
3

 using three equal subintervals with right endpoints, left endpoints, and midpoints.

x	3	4	5	6	7	8	9
f(x)	−3.6	−2.1	−0.7	0.2	0.8	1.3	1.7
(a) Estimate 9	f(x) dx
3

 using three equal subintervals with right endpoints.

R₃ =  

If the function is known to be an increasing function, can you say whether your estimate is less than or greater than the exact value of the integral?

(b) Estimate 9	f(x) dx
3

 using three equal subintervals with left endpoints.

L₃ =  

If the function is known to be an increasing function, can you say whether your estimate is less than or greater than the exact value of the integral?

(c) Estimate 9	f(x) dx
3

 using three equal subintervals with midpoints.

M₃ =  

If the function is known to be an increasing function, can you say whether your estimate is less than or greater than the exact value of the integral?

Answer 1

I've sketched it out for you here:

The red rectangles use the "right endpoints", the green rectangles use the "left endpoints" and the black dotted lines use the midpoints. Each rectangle is 2 units wide. Use geometry to calculate the area of each rectangle and add the three red's to get the right hand sum, add the three greens to get the left hand sum, etc.