$F(0)=\int_1^0f(t)dt$

The integral represents the area under the curve.
Notice that the bounds are reversed.

$F(1)=\int_1^1f(t)dt$

What is the value of an integral if the bounds are equal?

$F^\prime(3)=\frac{d}{dx}\int_1^3f(t)dt$

If you evaluate the integral above, will your answer be a number or an expression?
What is the derivative of the number or expression?

What is the area under the curve from $x = 1$ to $x = 6$?