### Home > APCALC > Chapter 1 > Lesson 1.2.4 > Problem1-73

1-73.

Sketch $f(x) = 3\sqrt { x + 1 }$ on $0 ≤ x ≤ 6$ three times, on three different sets of axes.

1. Review your work from problems 1-25 and 1-36. Use a similar process to approximate the area under the curve for $0 ≤ x ≤ 6$ using:

1. Six left endpoint rectangles.

The height of each rectangle is $1$. The bases can be determined by the function.

2. Six right endpoint rectangles.

The height of each trapezoid is $1$. The bases can be determined by the function.

3. Six trapezoids.

The base of each rectangle is $1$. The heights can be determined by the function.

2. Which approximations were overestimates of (greater than) the actual area? Which were underestimates? Explain.

$f(x)$ is an increasing and concave down function. Experiment with other increasing functions, and then experiment with other concave down functions. What do you notice about the under/over nature of left-endpoint, right-endpoint and trapezoidal approximations in these more general cases?

3. Which approximation is the most accurate? Explain.

Use the eTool below to view the graphs of the function.
Click the link at right for the full version of the eTool: Calc 1-72 HW eTool