Sketch a graph of the region bounded by the functions $f(x)=x^2,g(x)=−2x+8,$ and the $x$-axis.

1. How can you estimate the area in this region?

   Hint (a):

   Graph the functions. Shade the region. Find all relevant intersections: Where does $f(x)$ intersect with $g(x)$? Where does each function cross the $x$-axis? How many integrals will you need?

   More Help:

   The functions intersect at $(2, 4)$. The area should be approximated with rectangles or trapezoids for $0\le x\le2$ and added to the area of the triangle for $2\le x\le4$.

2. Using your method, estimate the area of the region.

   Answer (b):

   $\text{Estimates will vary. area }\approx \frac{1}{500}\displaystyle\sum_{n=0}^{999}\Big(\frac{n}{500}\Big)^2+\frac{1}{2}(2)(4)\approx 6.66\text{ units}^2$