Answer:
(4 , 0.25)
Step-by-step explanation:
M = (3.2 + 1.6 / 2 , 2.5 - 4.5 / 2 )
therefore, the midpoint is (4 , 0.25)
Answer:
96
Step-by-step explanation:
multiply by 12 because thats how many inches are in a foot
Formula for Riemann Sum is:
![\frac{b-a}{n} \sum_{i=1}^n f(a + i \frac{b-a}{n})](https://tex.z-dn.net/?f=%5Cfrac%7Bb-a%7D%7Bn%7D%20%5Csum_%7Bi%3D1%7D%5En%20f%28a%20%2B%20i%20%5Cfrac%7Bb-a%7D%7Bn%7D%29)
interval is [1,3] so a = 1, b = 3
f(x) = 3x , sub into Riemann sum
![\frac{2}{n} \sum_{i=1}^n 3(1 + \frac{2i}{n})](https://tex.z-dn.net/?f=%5Cfrac%7B2%7D%7Bn%7D%20%5Csum_%7Bi%3D1%7D%5En%203%281%20%2B%20%5Cfrac%7B2i%7D%7Bn%7D%29)
Continue by simplifying using properties of summations.
![= \frac{2}{n}\sum_{i=1}^n 3 + \frac{2}{n}\sum_{i=1}^n \frac{6i}{n} \\ \\ = \frac{6}{n}\sum_{i=1}^n 1 + \frac{12}{n^2}\sum_{i=1}^n i \\ \\ =\frac{6}{n} (n) + \frac{12}{n^2}(\frac{n(n+1)}{2}) \\ \\ =6+\frac{6}{n}(n+1) \\ \\ =12 + \frac{6}{n}](https://tex.z-dn.net/?f=%3D%20%5Cfrac%7B2%7D%7Bn%7D%5Csum_%7Bi%3D1%7D%5En%203%20%2B%20%20%5Cfrac%7B2%7D%7Bn%7D%5Csum_%7Bi%3D1%7D%5En%20%5Cfrac%7B6i%7D%7Bn%7D%20%5C%5C%20%20%5C%5C%20%3D%20%5Cfrac%7B6%7D%7Bn%7D%5Csum_%7Bi%3D1%7D%5En%201%20%2B%20%20%5Cfrac%7B12%7D%7Bn%5E2%7D%5Csum_%7Bi%3D1%7D%5En%20i%20%5C%5C%20%20%5C%5C%20%3D%5Cfrac%7B6%7D%7Bn%7D%20%28n%29%20%2B%20%5Cfrac%7B12%7D%7Bn%5E2%7D%28%5Cfrac%7Bn%28n%2B1%29%7D%7B2%7D%29%20%5C%5C%20%20%5C%5C%20%3D6%2B%5Cfrac%7B6%7D%7Bn%7D%28n%2B1%29%20%5C%5C%20%20%5C%5C%20%3D12%20%2B%20%5Cfrac%7B6%7D%7Bn%7D%20)
Now you have an expression for the summation in terms of 'n'.
Next, take the limit as n-> infinity.
The limit of
![\frac{6}{n}](https://tex.z-dn.net/?f=%5Cfrac%7B6%7D%7Bn%7D)
goes to 0, therefore the limit of the summation is 12.
The area under the curve from [1,3] is equal to limit of summation which is 12.
What but what’s even the question