Let f be a function that is continuous on the closed interval [1,7]. A differentiable function F satisfies F′(x)=f(x) for all x in [1,7] and is known to have the values F(1)=2 and F(7)=15.
(a) Explain, without performing any direct numerical approximation, why the definite integral ∫17f(x)dx can be evaluated using only the information provided about F. In your explanation, discuss the hypotheses that must be met and how they are satisfied in this setting.
(b) Determine the exact value of ∫17f(x)dx and clearly justify your result.
(c) Suppose instead the integral were written with reversed limits, ∫71f(x)dx. Describe how the value you found in part (b) changes and explain the reason for this change.