Question

Let f(x)=x^2f ( x ) = x 2. Find the Riemann sum for ff on the interval [0,2][ 0 , 2 ], using 4 subintervals of equal width and taking the sample points to be the left endpoints. (Round your answer to two decimal places.) Group of answer choices

Answer 1

`A_L=1.75`

Explanation:

We are given:

`f(x)=x^2`

`interval = [a,b] = [0,2]`

Since
`n = 4` ⇒
`\Delta x = (b-a)/(n) = (2-0)/(4)=(1)/(2)`

Riemann sum is area under the function given. And it is asked to find Riemann sum for the left endpoint.

`A_L= \sum\limits^(n)_(i=1)\Delta xf(x_i) = (1)/(2)(0^2+((1)/(2))^2+1^2+((3)/(2))^2)=(7)/(4)=1.75`

Note:

If it will be asked to find right endpoint too,

`A_R=\sum\limits^(n)_(i=1)\Delta xf(x_i) =(1)/(2)(((1)/(2))^2+1^2+((3)/(2))^2+2^2)=(15)/(4)=3.75`

The average of left and right endpoint Riemann sums will give approximate result of the area under
`f(x)=x^2` and it can be compared with the result of integral of the same function in the interval given.

So,
`(A_R+A_L)/2 = (1.75+3.75)/2=2.25`

`\int^2_0x^2dx=x^3/3|^2_0=8/3=2.67`

Result are close but not same, since one is approximate and one is exact; however, by increasing sample rates (subintervals), closer result to the exact value can be found.