Question

True or False? In Exercises 55-60, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.. The value of \(\int_{2}^{2} \sin \left(x^{2}\right) d x\) is \(0 .\)

Short answer

True. The value of \(\int_{2}^{2} \sin \left(x^{2}\right) dx\) is indeed \(0\) because the integral over an interval of length zero is always zero regardless of the function integrated.

Step-by-step solution

Understand the problem

In this exercise, the given integral is \(\int_{2}^{2} \sin \left(x^{2}\right) d x\). This integral represents the signed area under the curve \(y=\sin(x^2)\) between \(x=2\) and \(x=2\), which is a single point. Therefore, the integral over an interval of length zero is always zero regardless of the function integrated.

Evaluate the integral

Since the upper limit of the integral equals the lower limit, the integral evaluates to zero. Mathematically, an integral taken over a definite range where the upper limit equals to the lower limit will always be 0, regardless of the function. In other words, this is the property of definite integral: \(\int_{a}^{a} f(x) dx = 0\) for any function \(f(x)\).