Area of the Region Bounded Between Two Curves

Case (i): between two curves, integrating in $x$. Let $y=f(x)$ and $y=g(x)$ be curves with $f(x)\ge g(x)$ for all $x\in[a,b]$. The region bounded between them and the ordinates $x=a,x=b$ is divided into thin vertical strips of width $\Delta x$ and height $f(x)-g(x)\ge0$. Summing and passing to the limit,

Calling $y=f(x)$ the upper curve ($U$, viewed in the positive $y$-direction) and $y=g(x)$ the lower curve ($L$), this is written $A=\displaystyle\int_a^b(y_U-y_L)\,dx$.

Case (ii): between two curves, integrating in $y$. Let $x=f(y)$ and $x=g(y)$ with $f(y)\ge g(y)$ for all $y\in[c,d]$. Viewing in the positive $x$-direction, calling $x=f(y)$ the right curve ($R$) and $x=g(y)$ the left curve ($L$),

General working rule (no need to name curves globally as upper/lower). For a region bounded by $y=f_1(x)$, $y=f_2(x)$ and the lines $x=a,\,x=b$ ($a<b$): draw an arbitrary vertical line cutting the region; let $y_{\text{ENTRY}}$ be the $y$-value where the line enters the region and $y_{\text{EXIT}}$ where it exits (both read off the bounding curves' equations at that $x$). Then

The mirror rule (region bounded by $x=g_1(y),\,x=g_2(y)$ and $y=c,\,y=d$, $c<d$): draw a horizontal line, find $x_{\text{ENTRY}}$ and $x_{\text{EXIT}}$, and

Recurring worked patterns (Examples 9.47-9.61). The area of an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is $\pi ab$ (circle, $b=a$: $\pi a^2$); the area between the parabola $y^2=4ax$ and its latus rectum ($x=a$) is $\frac83a^2$ — Archimedes' formula, two-thirds the area of the enclosing rectangle (base = latus rectum, height = distance from vertex to the latus rectum), and equivalently four-thirds the area of the inscribed triangle, exactly the ratio proved in §9.1; a triangle's area can be found purely by integration by writing each side as a line and adding/subtracting the areas under them; and the region between a tangent and normal to a circle can be computed by either vertical or horizontal strips, giving the same value both ways — a useful self-check. …