# Calculate the integral int^4_ (0) (4*x -x^2)dx using Riemann Sum and a regular partition with in subintervals Since the problem does not specify what is the order of regular partition and what points are considered (leftpoints or rightpoints), the problem will be solved considering a regular partition of order 4 and left endpoints.

You need to evaluate the length of each subinterval such that:

`Delta x =...

Start your 48-hour free trial to unlock this answer and thousands more. Enjoy eNotes ad-free and cancel anytime.

Since the problem does not specify what is the order of regular partition and what points are considered (leftpoints or rightpoints), the problem will be solved considering a regular partition of order 4 and left endpoints.

You need to evaluate the length of each subinterval such that:

`Delta x = (4 - 0)/4 = 1`

You need to evaluate the left endpoints such that:

`x_0 = 0`

`x_1 = 0+1 = 1`

`x_2 = 1+1 = 2`

`x_3 = 2+1 = 3`

You need to evaluate the area under the curve `f(x) = 4x -x^2, x`  axis and the endpoints `x=0`  and `x=4`  using the 4 rectangles, such that:

`A_4 = f(x_0)*Delta x+ f(x_1)*Delta x + f(x_2)*Delta x + f(x_3)*Delta x`

`A_4 = f(0)*1 + f(1)*1 + f(2)*1 + f(3)*1`

`A_4 = (4*0 -0^2) + (4*1 -1^2) + (4*2 -2^2) + (4*3 -3^2)`

`A_4 = 3 + 4 + 3 = 10`

Hence, evaluating the definite integral using a regular partition of order 4 and left endpoints yields `int_0^4 (4x -x^2)dx~~ 10.`

Approved by eNotes Editorial Team