CPM Homework Help : CALC Problem 6-126

The graphs of the equations $x = y^2$ , $x = y −1$ , $y = 2$ , and $y =-2$ are shown below. These four graphs form the boundary of a region.

Show that the area of this region is ^{$\frac { 28 } { 3 }$}un^{$^{2}$}. Hint: Use horizontal rectangles.

Hint 1:

$dy$ :
Since we are using horizontal rectangles, the infinitely small widths of each rectangle are measured on the $y$ -axis...
hence we us $dy$ instead of $dx$ .

Hint 2:

The bounds:
Notice that BOTH equations are written with $y$ as the input and $x$ as the output.
Therefore, find the bounds of integration will be on the $y$ -axis.
$A =$ lower $y$ value
$B =$ upper $y$ value

Hint 3:

The integrand:
The 'top function' and 'bottom function' will also be determined with respect to the y-axis.
Let the 'top function' be the one with the right-most position, while the 'bottom function' will be the one with the 'left-most position.'
Another way to look at this is the 'top function' has the highest $x$ -values, while the 'bottom function' has the lower $x$ -values.