Area Between Two Curves- Integration Method Explained

What Is the Area Between Two Curves?

When you have two functions overlapping on a graph, the region trapped between them isn't always a simple shape. You can't just use basic geometry formulas. This is where integration comes in.

The area between two curves is found by integrating the difference between the top function and the bottom function over the interval where they intersect. That's the core idea. Everything else is just applying this concept correctly.

The Basic Formula

If f(x) stays above g(x) on the interval [a, b], the area is:

A = ∫[a,b] (top function − bottom function) dx

That's it. The hard part is figuring out which function is on top and where the functions actually intersect.

Step-by-Step: Finding the Area

Step 1: Find the Intersection Points

Set the two functions equal to each other and solve for x. These points mark where one function crosses the other. They're your limits of integration.

Step 2: Determine Which Function Is on Top

Pick a test point between your intersections. Plug it into both functions. Whichever gives the larger y-value is the top function.

This matters. Get it backwards and you'll get a negative answer. Take the absolute value if that happens, or just redo the setup.

Step 3: Set Up the Integral

Subtract the bottom function from the top function. Integrate with respect to x from the left intersection to the right intersection.

Step 4: Evaluate

Find the antiderivative, apply the limits, and subtract. Don't forget your constant of integration—it's zero in definite integrals, so it doesn't matter here.

Worked Example

Find the area between y = x² and y = x.

Step 1: Set them equal: x² = x → x² - x = 0 → x(x - 1) = 0

Intersections at x = 0 and x = 1.

Step 2: Test x = 0.5. f(0.5) = 0.25, g(0.5) = 0.5. The line y = x is higher on this interval.

Step 3: Set up the integral:

A = ∫₀¹ (x - x²) dx

Step 4: Evaluate:

A = [½x² - ⅓x³]₀¹ = (½ - ⅓) - (0) = ⅙

The area is 1/6 square units.

Horizontal Slices: When to Use dx vs dy

Sometimes it's easier to slice the region horizontally instead of vertically. This happens when the region is bounded on the left and right by functions of y rather than x.

The formula flips:

A = ∫[c,d] (right function − left function) dy

You integrate with respect to y, so your functions need to be written as x = f(y) and x = g(y). Solve your original equations for x first.

Example with Horizontal Slices

Find the area between x = y² and x = y + 2.

Solve for y: y² = y + 2 → y² - y - 2 = 0 → (y - 2)(y + 1) = 0

Intersections at y = -1 and y = 2.

On this interval, x = y + 2 is to the right of x = y².

A = ∫₋₁² [(y + 2) - y²] dy

A = [½y² + 2y - ⅓y³]₋₁²

A = (2 + 4 - 8/3) - (½ - 2 + ⅓) = 4.5

Area = 9/2 square units.

Comparing Integration Methods

Common Mistakes to Avoid

• Forgetting intersections: Always find where the curves meet first. Your integral limits come from these points, not from the problem's stated interval.
• Wrong function on top: This gives you a negative area. Double-check with a test point.
• Switching variables incorrectly: If you use dy, rewrite everything in terms of y. Don't mix x and y in the same integral.
• Skipping the absolute value: If your difference ends up negative, take the absolute value or redo the setup.

Multiple Regions

Sometimes the curves switch positions. The top function on one interval might be the bottom function on another.

When this happens, split the integral at each intersection point. Calculate each piece separately, then add the absolute values together.

Example: If f(x) > g(x) on [a, c] but g(x) > f(x) on [c, b], your total area is:

A = ∫ₐᵇ|f(x) - g(x)|dx = ∫ₐᶜ (f - g)dx + ∫ᵈᵇ (g - f)dx

Practical How-To Checklist

• ☐ Graph both functions (mentally or on paper)
• ☐ Find all intersection points
• ☐ Identify which function is on top/bottom on each interval
• ☐ Choose vertical or horizontal slices based on what makes setup easier
• ☐ Set up the integral with correct limits
• ☐ Evaluate and simplify

When Integration Gets Complicated

Some regions don't have clean antiderivatives. When you can't integrate exactly, use numerical methods like Simpson's rule or the trapezoidal rule to approximate the area. Your calculator's fnInt function works fine for this.