8.5 Finding the Area Between Curves Expressed as Functions of y

Learning objectives

—CHA-5.A—Calculate areas in the plane using the definite integral—Topic 8.5• CHA-5.A Calculate areas in the plane using the definite integral.• CHA-5.A.2 Areas of regions in the plane can be calculated using functions of either x or y.• Enduring understanding CHA-5: Definite integrals allow us to solve problems involving the accumulation of change in area or volume over an interval.

When boundaries are written as $x$ -functions of $y$ , use a horizontal slice of thickness $dy$ . If $x=r(y)$ is the right boundary and $x=\ell(y)$ is the left boundary from $y=c$ to $y=d$ , the slice width is $r(y)-\ell(y)$ .

A=\int_c^d\big(\text{right}(y)-\text{left}(y)\big),dy

Integration variable	Slice	Difference	Bounds
$dx$	vertical	upper $-$ lower	$x$ -values
$dy$	horizontal	right $-$ left	$y$ -values

For the region bounded by $x=y^2$ and $x=2y$ , intersections satisfy $y^2=2y$ , so $y=0$ and $y=2$ . On $0<y<2$ , $2y$ is to the right of $y^2$ . Thus $A=\int_0^2(2y-y^2)\,dy=[y^2-y^3/3]_0^2=4/3$ square units.

A $dy$ integral must use expressions in $y$ and $y$ -value limits; do not mix them with $x$ -bounds. If a horizontal line meets different right or left curves in different vertical ranges, split the integral at the $y$ -value where that boundary changes.

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—8.6 Finding the Area Between Curves That Intersect at More Than Two PointsRead