Question

(a) Use the definition given below with right endpoints to express the area under the curve $y = x^3$ from 0 to 1 as a limit. The area A of the region S that is bounded above by the graph of a continuous function $y = f(x)$, below by the x-axis, and on the sides by the lines $x = a$ and $x = b$ is the limit of the sum of the areas of approximating rectangles. $A = lim_{n o infty} R_n = lim_{n o infty} left[ f(x_1)Delta x + f(x_2)Delta x + dots + f(x_n)Delta x ight] = lim_{n o infty} sum_{i=1}^{n} f(x_i)Delta x$ (b) Use the following formula for the sum of cubes of the first $n$ integers to evaluate the limit in part (a). $1^3 + 2^3 + 3^3 + dots + n^3 = left[ frac{n(n+1)}{2} ight]^2$

          (a) Use the definition given below with right endpoints to express the area under the curve $y = x^3$ from 0 to 1 as a limit.
The area A of the region S that is bounded above by the graph of a continuous function $y = f(x)$, below by the x-axis, and on the sides by the lines $x = a$ and $x = b$ is the limit of the sum of the areas of approximating rectangles.
$A = lim_{n 	o infty} R_n = lim_{n 	o infty} left[ f(x_1)Delta x + f(x_2)Delta x + dots + f(x_n)Delta x 
ight] = lim_{n 	o infty} sum_{i=1}^{n} f(x_i)Delta x$
(b) Use the following formula for the sum of cubes of the first $n$ integers to evaluate the limit in part (a).
$1^3 + 2^3 + 3^3 + dots + n^3 = left[ frac{n(n+1)}{2} 
ight]^2$

$(a) Use the definition given below with right endpoints to express the area under the curve y = x^3 from 0 to 1 as a limit. The area A of the region S that is bounded above by the graph of a continuous function y = f(x), below by the x-axis, and on the sides by the lines x = a and x = b is the limit of the sum of the areas of approximating rectangles. A = limn o infty Rn = limn o infty left[ f(x1)Delta x + f(x2)Delta x + dots + f(xn)Delta x ight] = limn o infty sumi=1^n f(xi)Delta x (b) Use the following formula for the sum of cubes of the first n integers to evaluate the limit in part (a). 1^3 + 2^3 + 3^3 + dots + n^3 = left[ fracn(n+1)2 ight]^2$

Added by Cindy P.

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