### Home > APCALC > Chapter 2 > Lesson 2.3.1 > Problem2-108

2-108.

Given: $\left | x ^ { 2 } - 1 \right |$

1. Rewrite $h$ as a piecewise-defined function.

Notice how this $y = \left | x ^ { 2 } - 1 \right |$ graph differs from $y = x^2 − 1$. From this we can determine that there will be three pieces, and we already know the first and last piece.

The middle piece is a vertical reflection of $y = x^2 − 1$.

$x^2−1,x<−1$
$y=−x^2+1,−1\le x\le1$
$x^2 − 1, x > 1$

2. Using set notation, state the domain and range of $h$.

Consider how the range of $h(x)$ differs from the range of $y = x^2 − 1$.

3. Estimate the area under the curve for $−1\le x\le3$ by any method.

A trapezoidal sum is an option.

4. Write a Riemann sum to approximate the area under the curve for $−1\le x\le3$ using $24$ rectangles of equal width. Then evaluate the sum.

The height is determined by the function.

$\text{The width/base of each rectangle is }\frac{1}{6}.$

Use the eTool below to help answer each part:
Click on the link to the right to view the full version of the eTool. Calc 2-108 HW eTool