Area under curve

Question

Calculate 𝑀6 for 𝑓(𝑥)=√6x over [3,6].

Answer 1

I assume that M6 is the Riemann Sum using 6 equal subintervals and midpoints.

Δx = (6 - 3)/6 = 1/2

M6 = f(3.25)(1/2) + f(3.75)(1/2) + f(4.25)(1/2) + f(4.75)(1/2) + f(5.25)(1/2) + f(5.75)(1/2)

Answer 2

is X under the radical or not

case 1 : x is under the radical...

f(x) = sqrt(6x) = sqrt(6)sqrt(x) = sqrt(6) * x^(1/2)

the anti-derivative is sqrt(6) * (2/3)x^(3/2), x=3..6

(2/3)( 6^(3/2) - 3^(3/2)) = (2/3) ( 6*sqrt(6) - 3*sqrt(3))

= 4*sqrt(6) - 2 * sqrt(3)

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case 2: x is NOT under the radical...

f(x) = sqrt(6) * x

the anti-derivative is sqrt6) ( (1/2)x^2)

the limit is sqrt(6) [ 18 - 9/2] = sqrt(6)[27/2]