Max-Min Inequality Rule for Definite Integrals

Unit 8: Applications of Antiderivatives Max-Min Inequality Rule for Definite Integrals

The Max-Min Inequality Rule for Definite Integral is a very narrow rule, and therefore has a very narrow application. I have not seen this method used often, if at all, on the actual AP Calculus AB exam, and normally I see it brought up by AP Calc teachers for in class exams. It is included as a topic that the AP Calculus board wants you to know, so it is definitely fair game for the actual AP Calculus exam.

Definition: Max-Min Inequality Rule for Definite Integrals

If a function, $f (x)$ , has a maximum y-value , $f_{\max}$ , and a minimum y-value , $f_{\min}$ , on a closed x-interval , [ a , b ], then the Min-Max Inequality Rule applies,

$f_{\min} ∙ (b - a) \leq \int_{a}^{b} f (x) dx \leq f_{\max} ∙ (b - a)$

What it means:

When you cut through all the heavy math notation what the Min-Max Inequality Rule is saying is pretty straightforward. This is really an extension of Riemann Sums, here you are using only the area of a single rectangle ( Area = base x height ) to approximate the exact net area between the curve and the x-axis on the closed x-interval , [ a , b ] (Definite Integral).

What that math speak is saying is that there is the smallest rectangle you could make to approximate the definite integral, there is also the largest rectangle you could make to approximate the definite integral, and the exact value of the definite integral has to be somewhere between those two values.