Question

Hi it's me again
I need help with this too and I promise to never use this site again! I feel so ashamed for asking :(
Use the Fundamental Theorem to evaluate the integral from 0 to 3 of the quantity x cubed minus 6 times x, dx.The answer has to have the antiderivative too.
Also Use a graph of the function to explain the geometric meaning of the value of the integral.

Answer 1

you need to find F(x) such that dF/dx = x^3-6x
Using the power rule, that is just
F(x) = 1/4 x^4 - 3x^2 + c

Now, the definite integral is just

F(3)-F(0) = (1/4 * 81 - 3*9 + c) - (0-0+c)
= 81/4 - 27
= -27/4

Read your text on the meaning of the integral. Consider that it is a sum of many many very thin rectangles, of width dx and height f(x).