Mathematics and Statistics · Ch 7 — Applications of Definite Integration

Area with Respect to the y-axis

Sometimes a region is described more naturally in terms of the y-axis — bounded by a curve, the y-axis, and two horizontal lines $y=c$ and $y=d$ . In that case the roles of $x$ and $y$ are simply swapped in the area formula.

The formula

If the curve is written as $x=g(y)$ and the region is bounded by the curve, the y-axis, and the horizontal lines $y=c$ (lower) and $y=d$ (upper), with $x=g(y)\ge 0$ throughout, then the area is

A = \int_{c}^{d} x\,dy = \int_{c}^{d} g(y)\,dy

Here the thin strips are horizontal instead of vertical: a strip at height $y$ has length $x=g(y)$ and thickness $dy$ , so its area is $x\,dy$ , and integrating from $y=c$ to $y=d$ accumulates the whole region.

When to integrate with respect to $y$

The choice between $\int y\,dx$ and $\int x\,dy$ depends on which axis and which limits the region is described against:

Region bounded by	Integrate	Limits are values of
Curve, x-axis, and ordinates $x=a$ , $x=b$	$\displaystyle\int_{a}^{b} y\,dx$	$x$
Curve, y-axis, and lines $y=c$ , $y=d$	$\displaystyle\int_{c}^{d} x\,dy$	$y$

Definition 1Area with Respect to the y-axis

For a curve $x=g(y)$ bounded by the y-axis and the horizontal lines $y=c$ and $y=d$ , with $g(y)\ge0$ , the area is $A=\displaystyle\int_{c}^{d} x\,dy = \int_{c}^{d} g(y)\,dy$ — the same accumulation idea with …

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Area with Respect to the y-axis