Why Intersection Multiplicity Matters
When two curves meet, their order may change or stay the same. That distinction matters when calculating the area between them: the curve that is higher determines the integrand, so a change in order requires a new integral on the next interval.
A graph can give a useful first impression, but a point of contact can be difficult to classify by sight. The algebraic multiplicity of an intersection root provides a precise test. In particular, the parity of the multiplicity tells us whether the difference of the curves changes sign at that root.
Intersection Roots and Their Multiplicity
Let \(f\) and \(g\) be polynomial functions, and define their difference by \(D(x)=f(x)-g(x)\). The curves intersect at \(x=a\) exactly when \(D(a)=0\). The root \(a\) has multiplicity \(m\) if \(D(x)\) can be written as
where \(q\) is a polynomial. The condition \(q(a)\ne 0\) ensures that \(m\) is the highest power of \(x-a\) that divides \(D(x)\). A root of multiplicity one is simple; a root of multiplicity at least two is repeated.
Why Parity Predicts a Crossing
Near \(a\), the factor \(q(x)\) keeps the same sign as \(q(a)\). This is because \(q\) is continuous and \(q(a)\ne 0\), so sufficiently close to \(a\), \(q(x)\) remains nonzero and cannot change sign. The factor \((x-a)^m\) determines whether the sign of \(D\) changes as \(x\) passes through \(a\).
To justify the result, choose points just to the left and right of \(a\), close enough that \(q\) has the same nonzero sign at both points. The factor \(x-a\) is negative on the left and positive on the right. Raising these signs to an odd power preserves the sign change; raising them to an even power makes both powers positive. Multiplying by \(q(x)\), whose sign has stayed fixed, gives the stated behavior of \(D\).
This result classifies whether the curves switch order. There is a related geometric detail: if \(m\ge 2\), then \(D'(a)=0\), so \(f'(a)=g'(a)\) when the curves are differentiable. They have the same tangent slope. But a repeated root of odd multiplicity, such as multiplicity three, still changes sign and the curves still cross. Sharing a tangent slope does not by itself mean that the curves stay on the same side of each other.
When an Area Integral Needs a Split
On any interval where \(D\) has one fixed sign, the same curve stays above the other. The area formula can therefore use one consistent upper-minus-lower integrand throughout that interval. At an interior root of odd multiplicity, the sign changes, so the upper and lower curves switch places and the integral must be split there. An interior root of even multiplicity does not switch their order, so it does not require a split just to keep the integrand correctly ordered.
A root can still be relevant as an endpoint of a region or for describing where the curves meet. The decision here is narrower: when integrating across an interval, split at an interior intersection if the ordering changes there. At an endpoint, there is no interval on the other side within the region to consider.
Set \(D=f-g\). Its zeros are the intersection points.
Factor \(D\), and identify the power of each root factor.
Odd multiplicity means a sign change; even multiplicity means no sign change.
Split at interior roots where the sign changes, then identify the upper curve on each interval.
Worked Examples
Worked Example: An Even-Multiplicity Tangency
Find the area between \(f(x)=x^2\) and \(g(x)=0\) on \([-1,1]\), and decide whether the integral needs to be split at their intersection.
The difference is
The curves intersect at \(x=0\), and \(D(x)=(x-0)^2\), so the root has multiplicity two. This is even, so \(D\) does not change sign at zero. Indeed, \(x^2\ge 0\) on the entire interval, meaning \(f\) stays above \(g\). No split is needed.
The area is
Check: the derivative of \(x^3/3\) is \(x^2\), the integrand. Also, \(x^2\) is nonnegative throughout the interval, consistent with \(f\) being the upper curve and the area being positive.
Worked Example: An Odd-Multiplicity Crossing
Find the area between \(f(x)=x^3\) and \(g(x)=0\) on \([-1,1]\). Determine where the integral must be split.
Here \(D(x)=x^3=(x-0)^3\), so the intersection at zero has multiplicity three, which is odd. The sign changes: \(x^3<0\) when \(x<0\), and \(x^3>0\) when \(x>0\). Thus the curves cross at zero. On \([-1,0]\), \(g\) is above \(f\); on \([0,1]\), \(f\) is above \(g\). Split at zero and integrate upper minus lower on each interval:
The first integral is \(0-(-1/4)=1/4\). The second is \(1/4-0=1/4\). The total area is \(1/4+1/4=1/2\).
Check: both antiderivatives differentiate to their integrands, since \(\frac{d}{dx}(-x^4/4)=-x^3\) and \(\frac{d}{dx}(x^4/4)=x^3\). The signs on the two intervals confirm that each integrand is upper curve minus lower curve.
Worked Example: One Crossing and One Tangency
Let \(f(x)=(x-1)^2(x+2)\) and \(g(x)=0\) on \([-3,2]\). Find the area and determine which intersection requires a split.
The difference is already factored:
There is a root at \(x=-2\) of multiplicity one, so the sign changes there. There is a root at \(x=1\) of multiplicity two, so the sign does not change there. Because \((x-1)^2\) is positive except at \(x=1\), the sign away from the roots is determined by \(x+2\): \(D<0\) for \(x<-2\), and \(D>0\) for \(x>-2\). Therefore \(g\) is above \(f\) on \([-3,-2]\), while \(f\) is above \(g\) on \([-2,2]\). The tangency at \(1\) does not require another split.
For integration, expand the difference:
An antiderivative is \(F(x)=x^4/4-3x^2/2+2x\), since \(F'(x)=x^3-3x+2\). On the first interval, the upper-minus-lower integrand is \(-D\); on the second it is \(D\). Thus
The endpoint values are
Therefore the first area is \(-(-6-3/4)=27/4\), and the second is \(2-(-6)=8\). The total is \(27/4+8=59/4\).
Check: differentiating \(F\) gives the expanded difference \(x^3-3x+2\). The signs show that each piece uses upper minus lower, and the computed pieces \(27/4\) and \(8\) are both positive. The root at \(1\) lies inside the second interval, but the curves keep the same order on both sides of it.
A Common Mistake: Splitting at Every Intersection
Finding all intersections is necessary, but automatically splitting the integral at every intersection is not. For example, in the previous example, a split at \(x=1\) would be unnecessary: \(D\) is positive both just before and just after \(1\), so \(f\) remains above \(g\). The split at \(x=-2\), by contrast, is essential because \(D\) changes from negative to positive there.
The reverse mistake is also possible: treating every repeated root as a non-crossing. Multiplicity alone is not enough unless its parity is considered. A root of multiplicity three is repeated, but it is odd, so the curves cross. Use the sign test, not the word “repeated,” to decide whether the order switches.
Why This Classification Is Useful
Multiplicity turns an intersection calculation into information about the geometry around that intersection. It identifies not only where the curves meet, but whether their order changes there. That can prevent both unnecessary splits, which add work, and missing splits, which can make an area integral use the wrong curve as the upper boundary.
For polynomial differences, factoring often reveals the answer directly. If factoring is not enough to make the signs clear, test one point in each interval between consecutive roots. The sign at that point determines the order throughout the interval, because a continuous function cannot change sign there without another zero.
Check Your Understanding
For each question, focus on the difference of the two curves and its sign on either side of an intersection.
- What does it mean for \(a\) to be a root of multiplicity \(m\) of \(D=f-g\)?
- Do curves whose difference has a root of multiplicity four switch order at that root? Explain using the sign of the root factor.
- If \(D(x)=(x+1)^3(x-2)^2\), which intersection roots change the order of the curves?
- Why can two curves share a tangent slope at an intersection and still cross there?
- What additional check can you use to identify the upper curve on an interval between intersection points?
