5.1 Integration & Anti-derivatives
Area Estimate Under a Function
Question #1 (Easy): Estimating the Area Under a Function
All questions in this subsection ask for area estimation under the given curve using:
1) Left endpoints:
2) Right endpoints:
Then asks which estimate is lower, upper, and best estimate.
- Do not assume that left endpoints always give the lower estimate and the right endpoints give the upper
- Given the pattern of the curve, it can be either/or; so make sure to check.
- Midpoints always give the best estimate.
Estimate the area under the given graph of using five rectangles over the interval by:
1) Using the left endpoints
2) Using the right endpoints
3) Using the midpoints
4) Which one is the lower estimate, upper estimate, and the best estimate of the true area?
The question specifies the number of rectangles to use, so
This affects the width of each rectangles over the interval , so
Then . Now substitute these input values into the sum, and then calculate:
1) Using the left endpoints:
2) Using the right endpoints:
3) Using the midpoints:
a. The midpoints for each subinterval are:
b. So for ,
; For for ,
; and so forth.
4) As the estimates show, is the lower estimate, is the upper estimate, and is the best estimate.
The reason or this is also because the function is consistently increasing over the interval.