Topic 8.5 Notes – Finding the Area Between Curves Expressed as Functions of y

What Area Between Curves Using y Means

You already know the usual formula with vertical slices:

$$A = \int (\text{top} - \text{bottom})\, dx$$

That works when curves are written as $y = f(x)$.

Now suppose the curves are written as:

$$x = f(y) \quad \text{and} \quad x = g(y)$$

A horizontal slice has:

• Thickness $dy$
• Length = rightmost $x$ − leftmost $x$

So the area formula becomes:

$$A = \int (\text{right} - \text{left})\, dy$$

It’s the same idea. We’re still adding up thin rectangles. The only difference is the direction of the slices.

In the diagram below, the region between the two parabolas is shaded and a horizontal sample slice is drawn across it.

The geometry drives everything. Horizontal slices measure width, not height. That width is the distance from the left curve to the right curve at a given $y$.

Setting Up the Integral with Respect to y

Let’s walk through the structure you’ll use on a quiz or FRQ.

1. Find intersection points in terms of y

Set the equations equal:

$$f(y) = g(y)$$

Solve for y-values. These are your limits of integration.

If you’re integrating $dy$, your bounds must be y-numbers. That’s a common place students lose points.

2. Decide which function is right and which is left

For a horizontal slice:

• Larger $x$-value → right function
• Smaller $x$-value → left function

If it’s unclear, plug in a test $y$-value between the bounds and compare the resulting $x$-values. A quick sketch also saves mistakes.

3. Write the integral

$$A = \int_{y=a}^{y=b} \big(x_{\text{right}} - x_{\text{left}}\big)\, dy$$

Then integrate normally and apply the Fundamental Theorem of Calculus.

If you subtract in the wrong order, your answer comes out negative. The area should be positive, so that’s your red flag.

When Horizontal Slices Are Better

You’re allowed to choose either variable. The smart move is choosing the one that keeps the setup clean.

Horizontal slices are helpful when:

• The equations are already written as $x = f(y)$
• Solving for $y$ would introduce messy ± square roots
• Vertical slices would force you to split the region into multiple integrals

Here’s the comparison:

On AP problems, they often design the region so one method avoids splitting. If one setup looks messy, try flipping the variable.

Calculator vs Non-Calculator Situations

Non-calculator:

• Solve intersections algebraically.
• Integrate exactly.
• Give exact answers (fractions, radicals).

Calculator active:

• You might approximate intersection points.
• You may evaluate a definite integral numerically.
• You still must show the correct setup with right − left and correct y-bounds.

Even on calculator FRQs, most points come from the setup, not the decimal.

Common Mistakes

• Subtracting top − bottom out of habit instead of right − left.
• Using x-bounds when integrating $dy$.
• Forgetting to rewrite everything in terms of $y$. No stray $x$’s allowed inside the integral.
• Not splitting the integral if the right/left relationship changes over the interval.

If the region switches sides midway, you must break it into two integrals.