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The graphs of the equations , , , and  are shown at right. These four curves form the boundary of a region.

Show that the area of this region is .
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.

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

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

The integrand:
The 'top function' and 'bottom function' will also be determined with respect to the -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 -values, while the 'bottom function' has the lower -values.