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

Show that the area of this region is .
Hint: Use horizontal rectangles.


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

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

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.