Question

In Exercises \(1-4\), set up the definite integral that gives the area of the region. $$ \begin{array}{l} f(x)=x^{2}-6 x \\ g(x)=0 \end{array} $$

Short answer

The definite integral that gives the area of the region between the functions is \[\int_{0}^{6}|x^2 - 6x|dx.\]

Step-by-step solution

Find the Intersection Points

The intersection points of the two functions can be found by setting the two functions equal to each other and solving for \(x\).\n\(f(x) = g(x)\) gives us \(x^2 - 6x = 0\). Solving for \(x\), we find \(x = 0, 6\).

Setup the Definite Integral

The definite integral is set up with the intersection points as the boundaries and the absolute difference of the functions as the integrand.\nHence, the integral will be \[\int_{0}^{6}|f(x)-g(x)|dx\]. Since \(g(x)=0\), the absolute value of \(f(x)-g(x)\) is simply \(|f(x)|\), so the integral becomes \[\int_{0}^{6}|x^2 - 6x|dx.\]

Write the Final Result

We have now successfully set up the definite integral that gives the area of the region between the curves \(f(x)\) and \(g(x)\):\nThe final answer is \[\int_{0}^{6}|x^2 - 6x|dx.\]