Choose vertical strips when both curves are naturally written in terms of $x$, and horizontal stri

Question

Choose vertical strips when both curves are naturally written in terms of $x$, and horizontal strips when they are naturally written in terms of $y$. Find all intersections first and split wherever the outer curve changes.

Vertical strips: top minus bottom

Area in terms of x

\[A=\int_a^c[\mathrm{f}(x)-\mathrm{g}(x)]\,\mathrm{d}x+\int_c^b[\mathrm{g}(x)-\mathrm{f}(x)]\,\mathrm{d}x\]

Horizontal strips: right minus left

Area in terms of y

\[A=\int_p^q[x_{\rm right}(y)-x_{\rm left}(y)]\,\mathrm{d}y\]

Sketch and label the curves and their intersection points.
Choose vertical or horizontal strips to avoid unnecessary splitting.
On each interval, identify top minus bottom or right minus left.
Add the non-negative areas of the separate regions.

Accepted Answer

With respect to $x$: integrate $\text{top}-\text{bottom}$. With respect to $y$: integrate $\text{right}-\text{left}$.

Areas Between Curves

Areas Between Curves

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