8.5 Finding the Area Between Curves Expressed as Functions of y

Integrate with respect to y

When curves are easier to describe as x = g(y), slice horizontally instead of vertically. The area between two curves x = f(y) (right) and x = g(y) (left), from y = c to y = d, is A = ∫[c,d] (f(y) − g(y)) dy. Always subtract the right curve minus the left curve so the integrand stays positive.

Finding bounds and setting up

Solve each equation for x in terms of y, then find intersection points by setting the two x-expressions equal to get the y-limits c and d. Sketch the region to confirm which curve is farther right (larger x) across the interval — this determines the order of subtraction.

Why switch to y?

Some regions are bounded by curves that are functions of y but not x (like x = y² or sideways parabolas), or vertical slicing would require splitting the integral into multiple pieces. Rewriting in terms of y often gives a single, simpler integral instead of a sum of several vertical-strip integrals.

Key terms

Area between curves (in terms of y)

A = ∫[c,d] (f(y) − g(y)) dy, where f(y) is the right curve and g(y) is the left curve.

How to find limits c and d

Set the two x(y) expressions equal and solve for y — these are the intersection y-values.

When to integrate with respect to y

When curves are naturally functions of y, or when vertical slicing would require multiple integrals.

Determining which curve is 'right'

Plug in a test y-value between the bounds and compare the resulting x-values; the larger x is the right curve.

Horizontal strip width

The width of a horizontal rectangle at height y is f(y) − g(y), the right x minus the left x.

Try an AP-style question

What is the area between x = y² and x = 4?

1. 16/3
2. 64/3
3. 8
4. 32/3