Question

refer to the figure to the right. set up a definite integral from a to b that represents the shaded area.

Explanation:

Step1: Recall the formula for the area between a function and a horizontal line

The formula for the area between \(y = f(x)\) and \(y = c\) (where \(c\) is a constant) from \(x=a\) to \(x = b\) is \(\int_{a}^{b}(c - f(x))dx\). In this case, the horizontal line \(y = c\) (the upper - bound of the shaded region) is such that the height of the vertical rectangles (used in the Riemann sum definition of the integral) is \(c - f(x)\) (since \(c\geq f(x)\) for \(x\in[a,b]\)).

Step2: Set up the definite integral

Since the upper function is a constant (let's call the \(y\) - value of the horizontal line \(y = c\) and the lower function is \(y = f(x)\)), the definite integral representing the shaded area is \(\int_{a}^{b}(c - f(x))dx\).

Answer:

\(\int_{a}^{b}(c - f(x))dx\)