CPM Homework Help : APCALC Problem 1-121

WHICH IS BETTER? Part Two

Below is a comparison between using rectangles and trapezoids to approximate the area under a curve for the same interval of a function. Decide which method you think will best approximate the area under the curve for $a ≤ x ≤ b$ . Then approximate the area using each method if $f(x) = -0.25x(x - 9)$ , $a = 2$ , and $b = 8$ using $3$ sections. Compare your results with the actual area $A = 25.5 \text{ un}^2$ .

First quadrant graph, downward parabola, vertex in center of quadrant, with point at the origin, 3 equal width, vertical shaded bars, bottom edges on x axis, left edge of first bar labeled a, right edge of last bar labeled b, & with midpoint of top edge of each bar, on the parabola.

Midpoint Rectangles

First quadrant graph, downward parabola, vertex in center of quadrant, with point at the origin, 3 equal width vertical shaded trapezoids, bottom edges on x axis, left edge of first bar labeled a, right edge of last bar labeled b, & top right & left vertices of each bar is on the parabola, so each top edge is slanted, connecting consecutive x integer points on the parabola.

Trapezoids

Hint 1:

Which method comes closer to the actual curve $f(x)$ ?

Hint 2:

Each section is $2$ units wide. Use this knowledge and the equation of the function to find the areas of each section for both methods.

Hint 3:

Which method gave you the number closest to $25.5$ ?