Question

(a) Use the following definition to find an expression for the area under the curve $y = x^3$ from 0 to 2 as a limit. The area A of the region S that lies under the graph of the continuous function f is the limit of the sum of the areas of approximating rectangles: A = $lim_{n o infty} R_n = lim_{n o infty} [f(x_1)Delta x + f(x_2)Delta x + .. + f(x_n)Delta x]$ A = $lim_{n o infty} sum_{i=1}^{n} left(frac{2i}{n} ight)^3 cdot frac{2}{n}$ $lim_{n o infty} sum_{i=0}^{n} left(frac{2i}{n} ight)^3 cdot frac{2i}{n}$ $lim_{n o infty} sum_{i=2}^{n} left(frac{2i}{n} ight) cdot frac{2}{n}$ $lim_{n o infty} sum_{i=1}^{n} left(frac{2i}{n} ight)^3 cdot frac{2i}{n}$ $lim_{n o infty} sum_{i=2}^{n} left(frac{2i}{n} ight)^3 cdot frac{2}{n}$ (b) Use the following formula to evaluate the limit in part (a). $1^3 + 2^3 + 3^3 + ... + n^3 = left[frac{n(n+1)}{2} ight]^2$

          (a) Use the following definition to find an expression for the area under the curve $y = x^3$ from 0 to 2 as a limit.
The area A of the region S that lies under the graph of the continuous function f is the limit of the sum of the areas of approximating rectangles:
A = $lim_{n 	o infty} R_n = lim_{n 	o infty} [f(x_1)Delta x + f(x_2)Delta x + .. + f(x_n)Delta x]$ 
A = $lim_{n 	o infty} sum_{i=1}^{n} left(frac{2i}{n}
ight)^3 cdot frac{2}{n}$ 
$lim_{n 	o infty} sum_{i=0}^{n} left(frac{2i}{n}
ight)^3 cdot frac{2i}{n}$ 
$lim_{n 	o infty} sum_{i=2}^{n} left(frac{2i}{n}
ight) cdot frac{2}{n}$ 
$lim_{n 	o infty} sum_{i=1}^{n} left(frac{2i}{n}
ight)^3 cdot frac{2i}{n}$ 
$lim_{n 	o infty} sum_{i=2}^{n} left(frac{2i}{n}
ight)^3 cdot frac{2}{n}$ 
(b) Use the following formula to evaluate the limit in part (a).
$1^3 + 2^3 + 3^3 + ... + n^3 = left[frac{n(n+1)}{2}
ight]^2$

$(a) Use the following definition to find an expression for the area under the curve y = x^3 from 0 to 2 as a limit. The area A of the region S that lies under the graph of the continuous function f is the limit of the sum of the areas of approximating rectangles: A = limn o infty Rn = limn o infty [f(x1)Delta x + f(x2)Delta x + .. + f(xn)Delta x] A = limn o infty sumi=1^n left(frac2in ight)^3 cdot frac2n limn o infty sumi=0^n left(frac2in ight)^3 cdot frac2in limn o infty sumi=2^n left(frac2in ight) cdot frac2n limn o infty sumi=1^n left(frac2in ight)^3 cdot frac2in limn o infty sumi=2^n left(frac2in ight)^3 cdot frac2n (b) Use the following formula to evaluate the limit in part (a). 1^3 + 2^3 + 3^3 + ... + n^3 = left[fracn(n+1)2 ight]^2$

Added by Adam O.

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