Complementary Areas Under Inverse Functions

Why a Function and Its Inverse Share a Rectangle

A strictly increasing function and its inverse describe the same curve with the horizontal and vertical coordinates exchanged. That coordinate swap has a useful geometric consequence: between the endpoints of the curve, the area associated with the function and the area associated with its inverse fit together to fill a rectangle.

The key is to measure each area from the sides of that rectangle, not automatically from the coordinate axes. This distinction matters whenever the rectangle’s lower or left side is not an axis.

The Setting and the Two Areas

Let \(f:[a,b]\to\mathbb{R}\) be continuous and strictly increasing, with \(a<b\). Strict increase means that \(x_1<x_2\) implies \(f(x_1)<f(x_2)\). It ensures that \(f\) takes each value in its range exactly once, so its inverse \(f^{-1}\) is defined on \([f(a),f(b)]\). Continuity ensures that the graph and the inverse graph have no jumps on these intervals.

The graph of \(f\) runs from \((a,f(a))\) to \((b,f(b))\). These points are opposite corners of a rectangle with horizontal range from \(a\) to \(b\), and vertical range from \(f(a)\) to \(f(b)\). Its width is \(b-a\), its height is \(f(b)-f(a)\), and its area is their product.

For \(A\), a vertical slice at \(x\) runs from the rectangle’s bottom, \(y=f(a)\), up to \(y=f(x)\). For \(B\), a horizontal slice at \(y\) runs from the rectangle’s left side, \(x=a\), to the inverse curve, \(x=f^{-1}(y)\). The integrands are nonnegative because \(f\) and \(f^{-1}\) are increasing.

The Complementary-Area Theorem

The result uses areas measured from the rectangle’s own bottom and left sides. It does not claim that integrals of \(f\) and \(f^{-1}\), each measured from a coordinate axis, automatically add to the rectangle’s area.

Why the Areas Are Complementary

Consider any point \((x,y)\) inside the rectangle. If it lies below the graph, then \(f(a)\leq y\leq f(x)\), so it belongs to the region counted by \(A\). If it lies above the graph, then \(y>f(x)\). Since \(f\) is strictly increasing, its inverse preserves order, and this inequality is equivalent to \(f^{-1}(y)>x\). Thus the point lies to the left of the inverse curve, in the region counted by \(B\).

Every point in the rectangle is therefore in one of these two regions, except points on the graph itself. The two regions have no overlapping interiors. The curve separating them has zero area, so the areas of the two regions sum to the area of the rectangle. Using the definite-integral formulas for their areas gives the theorem.

This is a geometric proof: vertical slices measure one part of the rectangle, while horizontal slices measure the other. The inverse is essential because \(f^{-1}(y)\) gives the horizontal coordinate of the curve at height \(y\).

Worked Examples

A Common Pitfall: Forgetting the Rectangle’s Sides

A tempting but incorrect approach is to use \(\int_a^b f(x)\,dx\) as the first area and \(\int_{f(a)}^{f(b)} f^{-1}(y)\,dy\) as the second, then expect their sum to equal the rectangle’s area. These integrals measure from the coordinate axes, which need not be the rectangle’s bottom and left sides.

For \(f(x)=x^2\) on \([1,3]\), that incorrect calculation gives

not the rectangle’s area \(16\). The correction is to subtract the baselines \(f(a)=1\) and \(a=1\): the correct areas are \(\int_1^3(x^2-1)\,dx=\frac{20}{3}\) and \(\int_1^9(\sqrt{y}-1)\,dy=\frac{28}{3}\). These sum to \(16\). Always identify the rectangle’s bottom and left sides before writing the area integrals.

When the Identity Is Useful

The theorem can save work when one of the two areas is easier to calculate or is already known. Rather than integrating a complicated inverse function, find the rectangle’s area and subtract the known complementary area. It also provides a check on an area calculation: if the two areas do not sum to the rectangle’s width times height, revisit the bounds, the inverse, or the baseline subtractions.

The assumptions matter. Strict increase gives a single-valued inverse on the range, and continuity ensures that the graph and the inverse curve describe the regions without jumps. If the function is not strictly increasing, it may not have an inverse on the whole interval, so this version of the theorem cannot be applied directly.