Question

In Exercises 31-36, evaluate the integral using the following values. $$\int_{2}^{4} x^{3} d x=60, \quad \int_{2}^{4} x d x=6, \quad \int_{2}^{4} d x=2$$ $$ \int_{4}^{2} x d x $$

Short answer

-6

Step-by-step solution

Using the Property of Definite Integrals

The important thing to understand is that if we have an integral \(\int_{a}^{b} f(x) dx\), where \(b\) is greater than \(a\), and we reverse the limits of integration, we get \(\int_{b}^{a} f(x) dx = -\int_{a}^{b} f(x) dx\). This can be remembered as 'the reversal of the limits of integration changes the sign of the integral.' Applying this to \(\int_{4}^{2} x dx\), we find that it is equal to \(-\int_{2}^{4} x dx\).

Substituting the Given Value

We are given \(\int_{2}^{4} x dx = 6\) from the problem. To find the value of \(\int_{4}^{2} x dx\) by substituting this into our previous equation, \(\int_{4}^{2} x dx = -\int_{2}^{4} x dx = -6\).