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Question

(a) Use the following definition to find an expression for the area under the curve y=x^3 from 0 to 3 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→∞ Rn=lim n→∞ [f(x1)Δx+f(x2)Δx+⋯+f(xn)Δx]

(b) Use the following formula to evaluate the limit in part (a).

 1^3+2^3+3^3+⋯+n^3=[n(n+1)/2]^2

Answer 1

Answer in video because it cannot be pasted into this space.

Answer 2

a)

Δx = (3-0)/n

f(x_i) = f(iΔx) = (3i/n)³

R_n=Δx∑_i=1ⁿf(iΔx) = (3/n)∑_i=1ⁿ(3i/n)³ = (3⁴/n⁴)∑_i=1ⁿ i³

b)

R_n=(3⁴/n⁴)((n(n+1)/2)²)= (3⁴/(4n⁴)(n²(n²+2n+1))=(3⁴/4)(n⁴+2n³+n²)/n⁴=(3⁴/4)(1+ 2/n +1/n²)

A=lim n→∞ R_n = (3⁴/4)(1+0+0) = 3⁴/4 = 81/4