This is along calculus question, I didn\'t understand them, thanks! (1 pt) Defin

Question

This is along calculus question, I didn't understand them, thanks!

(1 pt) Definition: 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 = lim = lim (a) Use the above definition to determine which of the following expressions represents the area under the graph of f(x) from x -0 to x-2 64 n O A. lim O B. lim OC. lim 6 64 5 6 O D. lim 6 (b) Evaluate the limit that is the correct answer to part (a)·You may find the following formula helpful: 15 + 25 + 35 + + n5- 12 Value of limit

Explanation / Answer

Given f(x) = x5

a=0 , b=2

x = (b-a)/n = (2-0)n = 2/n

xi = a+i(x) = 0+i(2/n)

xi = i(2/n)

area = lim n-> i=1 to n f(xi) x

=lim n-> i=1 to n (i(2/n))5 (2/n)

=lim n-> (2/n)6 i=1 to n i5

=lim n-> (64/n6)i=1 to n i5

b)lim n-> (64/n6)i=1 to n i5

=lim n-> (64/n6) [n2(n+1)2(2n2+2n-1)]/12

=lim n-> (64/n4) [(n+1)2(2n2+2n-1)]/12

=lim n-> (64) [((n+1)/n)2((2n2+2n-1)/n2)]/12

=lim n-> (64) [(1+(1/n))2(2+(2/n) -(1/n2))]/12

=(64) [(1+(0))2(2+(0) -(02))]/12

=(64) [(1)(2)]/12

=64/6

=32/3

lim n-> (64/n6)i=1 to n i5 = 32/3

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