Question

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If the graphs of \(f\) and \(g\) intersect midway between \(x=a\) and \(x=b,\) then \(\int_{a}^{b}[f(x)-g(x)] d x=0\)

Short answer

The statement is false. This is because the intersection point doesn't necessarily split the two functions, f(x) and g(x), into equal areas between x=a and x=b.

Step-by-step solution

Understanding the statement

The statement is saying that if the graphs of functions f and g intersect midway between \(x=a\) and \(x=b\), then the integral from a to b of the difference \([f(x)-g(x)]\) is equal to 0. The scenario described here implies that the positive and negative areas under the graph between a and b cancel each other out, resulting in a net area of 0.

Evaluating the statement

While it's true that the area enclosed between two curves is given by the definite integral of the absolute difference between them, the statement still requires some considerations. The fact that \(f(x)\) and \(g(x)\) intersect midway doesn't guarantee equal area on both sides of the intersection point. This is because it's the shape of the functions, not the intersection point, that dictates the area underneath them.

Providing a counterexample

Consider the case where \(f(x) = x^2\) and \(g(x) = x\). They intersect at \(x=0\) and \(x=1\), which is midway between -1 and 1. However, \(\int_{-1}^{1}(x^{2}-x) dx\) clearly does not yield 0, specifically, it yields -\(\frac{1}{3}\). Thus, the statement is shown to be false.