1.3: Write the height h of the red rectangle as a function of x.

Step 1: Since they want the height defined in terms of x, what is the x-value of the rectangle?

Is there an exact value of x? No. There are a range of x-values here…

We need to write the function in terms of x, therefore look at the lowest possible x-value and the highest possible x-value. This is shown on the image (left) by the dotted red lines. The rectangle has an x-value somewhere between 0 and 2, or .

Step 2: Now find the height of the rectangle for any x-value between (and including) 0 and 2…

The height of this rectangle can be found by taking the height (e.g., distance from the x-axis) of the top function minus the height of the lower function.

In pictures:

	-		=
Height of the top function	-	Height of the bottom function	=	Height of the shaded region is the height of any rectangle whose x-value is between 0 and 2.

Therefore, the above exercise results in…

top – bottom

when and .

Substitute and get:

Test this: Compare your "math" result to what you can infer from the picture.

• What if the rectangle is at x = 0?
  
  This rectangle is really a point and we know that a point doesn't have height, so this rectangle should have a height equal to zero.
  
  Let x = 0 for .
  Thus, = 0.
  The math confirms that the height of the rectangle is zero!
• At x = 1, the height looks like 3. So, !
  A glimpse at the graph suggests that at x = 1 the height of the shaded region is 3, and the math confirms this!

Therefore, if given any x value between (and including) 0 and 2, we can now find the exact height of the shaded region!