# Set up and then use limits and the formula:Σ(superscript n)(subscript i=1) i^2 =(1/6)n(n + 1)(2n + 1)Set up and then use limits and the formula:Σ(superscript n)(subscript i=1) i^2 =(1/6)n(n +...

Set up and then use limits and the formula:Σ(superscript n)(subscript i=1) i^2 =(1/6)n(n + 1)(2n + 1)

Set up and then use limits and the formula:Σ(superscript n)(subscript i=1) i^2 =(1/6)n(n + 1)(2n + 1)  to find the exact value of ʃ^(superscript 2)(subscript 0) 3x^2dx

embizze | Certified Educator

Evaluate `int_0^2 3x^2dx` using Riemann sums:

Using a regular partition we get `Delta_(x_i)=(2-0)/n=2/n`

Using right endpoints we get `x_i^"*"=(2i)/n`

The Riemann sum is `sum_(i=1)^n f(x_i^"*")Deltax_i^"*"` so, taking the limit as `n->oo` we have:

`int_0^2 3x^2dx=lim_(n->oo) sum_(i=1)^n (3((2i)/n)^2)2/n`

Factoring out constants we get

`=lim_(n->oo) 24/n^3 sum_(i=1)^n i^2 `

** `(3((2i)/n)^2)2/n=(3)((4i^2)/n^2)(2/n)=24/n^3*i^2` ; n is a constant **

`=lim_(n->oo) [(24/n^3)((n(n+1)(2n+1))/6)]`

`=lim_(n->oo) [(24/n^3)(n^3/3+n^2/2+n/6)]` ** `lim cf(x)=c lim f(x)` **

`=24lim_(n->oo)[1/3+1/(2n)+1/(6n^2)]`  ** `lim_(n->oo) 1/n,1/n^2=0` **

`=24(1/3)=8`

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`int_0^2 3x^2dx=8`

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