Let $F$ be the equation $y=8-x$, let $G$ be the equation $y=\ln (x-7)$, and let $H$ be the equation $y=8$. Find the area of the region enclosed by the graphs of these equations.
(Express numbers in exact form. Use symbolic notation and fractions where needed.)
Answer
Finally, the area of the region enclosed by the graphs of these equations is given by the definite integral of the absolute difference of the functions over the interval from the smallest to the largest x-value of the points of intersection, which is \(\boxed{\int_{0}^{8} |x + \ln(x - 7) - 8| dx + \int_{8}^{7 + e^{8}} |\ln(x - 7) - 8| dx}\)
Step 1 ：Let's consider the three equations: \(F: y = 8 - x\), \(G: y = \ln(x - 7)\), and \(H: y = 8\). We are asked to find the area of the region enclosed by the graphs of these equations.
Step 2 ：To find the area enclosed by the graphs, we need to integrate the absolute difference of the functions over the interval where they intersect.
Step 3 ：First, we need to find the points of intersection of the three functions. This can be done by setting the functions equal to each other and solving for x.
Step 4 ：The points of intersection are \(x = 0\), \(x = 8\), and \(x = 7 + e^{8}\).
Step 5 ：Next, we integrate the absolute difference of the functions over the interval from the smallest to the largest x-value of the points of intersection.
Step 6 ：The area is then given by the definite integral of the absolute difference of the functions over this interval.
Step 7 ：Finally, the area of the region enclosed by the graphs of these equations is given by the definite integral of the absolute difference of the functions over the interval from the smallest to the largest x-value of the points of intersection, which is \(\boxed{\int_{0}^{8} |x + \ln(x - 7) - 8| dx + \int_{8}^{7 + e^{8}} |\ln(x - 7) - 8| dx}\)
