Question

1. tim wants to approximate the area underneath the curve y = 0.005x² - 2x + 200 for 0 ≤ x ≤ 200, shown shaded in the graph below.

he finds an initial estimate, a, for the shaded area by using fg and computing a = 1/2(200 units)(200 units)=20,000 square units.
the area of the shaded region is:
f. less than 20,000 square units, because the curve lies under fg.
g. less than 20,000 square units, because the curve lies over fg.
h. equal to 20,000 square units.
j. greater than 20,000 square units, because the curve lies under fg.
k. greater than 20,000 square units, because the curve lies over fg.

Explanation:

The initial estimate of the area is the area of the triangle formed by points \(F\), \(G\) and the origin. Since the curve \(y = 0.005x^{2}-2x + 200\) lies under the line segment \(FG\) for \(0\leq x\leq200\), the actual area under the curve (the shaded region) is less than the area of the triangle. The area of the triangle is calculated as \(\frac{1}{2}(200)(200)=20000\) square units. So the area of the shaded region is less than 20000 square units.

Answer:

F. less than 20,000 square units, because the curve lies under \(FG\).