Definite Integral Questions
I have a few questions that deal with definite integrals that I am confused about.
1) (a) Find the Riemann sum for f(x)=x^3, 2≤x≤12, if the partition points are 2,5,9,12 and the sample points are 3,8,10.
I found the answer for this part to be 5641.
I am stuck on part b:
(b) Find the Riemann sum if the partition points are 2,5,9,12 and the sample points are the midpoints.
2) A table of values of a decreasing function ff is shown. Use the table to find a lower and an upper estimate for the definite integral from 8 to -10 of f(x) dx. There is a chart & I will do my best to relay it.
x= -10 -7 -4 -1 2 5 8
f(x)= 9 5 2 -3 -4 -8 -15
3) The following sum
sin(10+(5/n)⋅(5/n) + sin(10+(5/n)⋅(5/n) + ... + sin(10+(5/n)⋅(5/n) is a right Riemann sum for the definite integral of b to 10 (b being the upper number) f(x) dx.
I found b to be 15 & f(x) to be sin(x).
The second part of the problem asks...
It is also a Riemann sum for the definite integral c to 0 (c being the top number) g(x) dx.
Where c equals?
I found this to be 5.
and g(x) equals?
Update: As I was working through this one again I found the correct answer to be sin(10 + x) but I'm uncertain as to why. Could someone give me an explanation of this?
4) Given that 7 ≤ f(x) ≤ 8 for -5 ≤ x ≤ 5, estimate the value of ∫ from 5 to -5 of f(x) dx.
The answer has two blanks to put the answer in...
__ ≤ ∫ from 5 to -5 of f(x) dx ≤ __
Then the last question I have seems to be practically the same concept...
Use property 8 to estimate the value of ∫ from 11 to 2 of 6/x dx.
__ ≤ ∫ from 11 to 2 of 6/x dx ≤ __