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

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

   Hint (a):

   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.

   Hint (a):

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

   Answer (a):

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

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

   Hint (b):

   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.

   Hint (c):

   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.

   Hint (d):

   The height is determined by the function.

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