CPM Homework Help : CALC3RD Problem 6-119

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

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

Sideways parabola facing right, vertex at the origin, intersects parallel lines at, y = negative 2 &, y = 2, at the points (4, comma 2), & (4 comma negative 2), increasing line passing through the points (negative 1, comma 0) & (0, comma 1), intersecting parallel lines at (1, comma 1), & (negative 3, comma negative 2), region inside is shaded.

Hint 1:

$\text{Area between curves}=\int_{A}^{B}((\text{top function})-(\text{bottom function}))dy$

Since we are using horizontal rectangles, the infinitely small widths of each rectangle are measured on the $y$ -axis, use $dy$ 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.