In the integration of `int_1^3 (x^2 + x)dx` using Riemann sum, what is the length of each subinterval.

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The area found under the curve `x^2 + x ` and between the limits 1 and 3 may be calculated using Riemann Sum.

S = `sum`  f(x)*`Delta x`

`f(x) = x^2 + x`

`sum f(x)*Delta x = sum x^2*Delta x + sum x*Delta x`

The interval [1,3] is divided into n subintervals. The length of each subinterval is `Delta x = (3-1)/n = 2/n`

The area under the curve f(x) is divided into n rectangles. The width of each rectangle is of  `2/n`  and the height is `f(x)=(x^2+x).`

`S = [(2/n)*(2/n)^2 + (2/n)*(2*2/n)^2 + ... + (2/n)*(2*k/n)^2 + ...+ (2/n)*(2*n/n)^2] + [(2/n)*(2/n) + (2/n)*(2*2/n) + ... + (2/n)*(2*k/n) +...+ (2/n)*(2*n/n)]`

`S = (2/n)^3*(1^2 + 2^2 + ... + n^2) + (2/n)^2*(1 + 2 + ... + n)`

`S = (8/n)^3*[n*(n+1)*(2n+1)/6] + (4/n^2)*[n*(n+1)/2]`

S = `lim_(n-gtoo)` `8*(2n^3 + 3n^2 + n)/(6n^3) ` + `lim_(n->oo)` `4*(n^2 + n)/(2n^2)`

S = `lim_(n->oo)` 8n^3*(2 + 3/n + 1/n^2)/6n^3 + `lim_(n->oo)` 4n^2(1+1/n)/2n^2

`S = 8/3 + 4/2 =gt S = 8/3 + 2 =gt S = 14/3`

If you choose to split the interval [1,3] into 5 subintervals (n=5), the length of each interval is: `Delta x = (3-1)/5 = 2/5`  = 0.4

The endpoints of each subintervals are: 1 ; 1.4 ; 1.8 ; 2.2 ; 2.6 ; 3.

Area estimation: `A = 1*f(1) + 1.4*f(1.4) + 1.8*f(1.8) + 2.2*f(2.2) + 2.6*f(2.6) + 3*f(3)`

ANSWER: The length of each subinterval, when [1;3] is divided into n subintervals is `Delta x = 2/n`  and the area is`int (x^2+x)dx = 14/3.`

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