Calculate the integral int^4_ (0) (4*x -x^2)dx using Riemann Sum and a regular partition with in subintervals

Expert Answers
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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.`