Find the area of the region bounded by the graphs of the given equations.
\[
y=5 x-x^{2}, y=x
\]
The area of the region is (Type an integer or a simplified fraction.)
Answer
The area of the region bounded by the graphs of the given equations is \(\frac{32}{3}\).
Step 1 ：First, we need to find the points of intersection of the two curves. This can be done by setting the two equations equal to each other and solving for x.
Step 2 ：Then, we integrate the absolute difference of the two functions over the interval defined by the points of intersection.
Step 3 ：Finally, we evaluate the integral to find the area. The integral represents the area of the region bounded by the two curves.
Step 4 ：The area of the region bounded by the graphs of the given equations is \(\frac{32}{3}\).
