Help with Definite Integral

Question

Write the Riemann sum to find the area under the graph of the function f(x) = x³ from x = 2 to x = 5.

options:

1.
2)
3)
4)

Answer 1

the anti-derivative is (1/4)x^4

limit as x-->5 : (1/4)(625)

limit as x-->2 : (1/4)(16) = 4

the area is 625/4 - 4 = 625/4 - 16/4 = 609/4 = 152.25

the Riemann sum:

lim deltaX-->0 SUM { deltaX* [f(Xi-1) - f(Xi)] } i=1,2,3,4,...n

for some partition {x0,x1,x2,....,xn}