Let
𝐴=∫1021𝑥d𝑥
and let 𝐿4
and 𝑅4
be the Left hand and Right hand Riemann sum approximations for 𝐴
with 4 partitions.
Which of the following is true?
𝐴<𝐿4<𝑅4
𝑅4<𝐴<𝐿4
𝐴<𝑅4<𝐿4
𝐿4<𝐴<𝑅4
𝐴
, 𝐿4
and 𝑅4
are all equal in this example
We can't determine the order of 𝐴
, 𝐿4
and 𝑅4
in this example
Calculate the value of the Right hand Riemann sum with 4 partitions.
Write your answer to 2 decimal places.
The interval of integration is [1, 2] and we have 4 partitions of equal width:
∆𝑥=2−11⁄4=1⁄4
The right endpoints of the intervals are: 5/4, 6/4, 7/4, 8/4.
Thus, the Riemann sum is:
𝑅4=∆𝑥(𝑓(5/4)+𝑓(6/4)+𝑓(7/4)+𝑓(8/4))
=1⁄4(2+3+4+5)
=3.5
Therefore, 𝑅4=3.5.
The answer is 𝑅4<𝐴<𝐿4. We can't determine the exact values of 𝐴, 𝐿4, and 𝑅4 without knowing the function being integrated, but we know that 𝑅4 is less than 𝐴 since the function is increasing on [1, 2]. We also know that 𝐿4 is greater than 𝑅4 because the function is increasing.