Given: $h ( x ) = | x ^ { 2 } - 1 |$  

1. Rewrite $h(x)$ as a piecewise function.

   Hint (a):

   Notice how this $y = |x^² − 1|$ 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.

   More Help (a):

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

   Answer (a):

   $x^² − 1, x < −1$
   $y = −x^² + 1, −1 ≤ x ≤ 1$
   $x^² − 1, x > 1$

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

   Hint (b):

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

3. Estimate $A(h, −1 ≤ x ≤ 3)$ by any method.

   Hint (c):

   A trapezoidal sum is an option.

4. Write a Riemann sum to approximate $A(h, −1 ≤ x ≤ 3)$ with $24$ rectangles of equal width and evaluate the sum.

   Hint (d):

   The height is determined by the function.