Find the values of C guaranteed by the mean value theorem for integrals.

Question

f(x)=x^3

[0,3]

Answer 1

well the integral is
x^4/4 + C

Now in between x = 0 and x = 3 the slope of the integral must be
f(0) = 0
f(3) = 27

so the integral must be between
0 + 0(3) = 0
or
0 + 27*3 = 81

In fact the integral is 3^4/4 = 81/4 or about 20