When Curves Meet, Do They Change Order?
Two curves can meet and then separate in different ways. They might pass through one another, changing which curve is above. Or they might touch and turn away while keeping the same order. That distinction matters when setting up an area integral: the curve order determines which function gives the height of each vertical slice.
A point where two curves meet is therefore not automatically a point where an area integral must be split. First determine whether the curves change order there.
Contact, Tangency, and Curve Order
Let \(f\) and \(g\) be continuous functions on an interval, and define their difference by \(D(x)=f(x)-g(x)\). The curves meet at \(x=c\) when \(f(c)=g(c)\), or equivalently \(D(c)=0\). For \(x\) near \(c\), the sign of \(D(x)\) tells us their order:
- If \(D(x)>0\), then \(f(x)>g(x)\), so \(f\) is above \(g\).
- If \(D(x)<0\), then \(g(x)>f(x)\), so \(g\) is above \(f\).
When discussing a tangency of differentiable graph curves at an interior point, we mean that the curves meet and have the same tangent slope there: \(f(c)=g(c)\) and \(f'(c)=g'(c)\). A touch without crossing has this slope equality when both functions are differentiable: their difference has a local maximum or minimum at \(c\), so its derivative there is zero. But equal slopes alone do not establish that the curves do not cross.
Why the Difference Determines the Area Setup
On a vertical slice at \(x\), the height between the graphs is the upper \(y\)-value minus the lower \(y\)-value. Since \(D(x)=f(x)-g(x)\), this height is \(|D(x)|\). Thus, over an interval \([a,b]\), the area between the graphs is
This formula also explains when a contact point matters. If \(D\) stays nonnegative throughout the interval, then \(f\) stays above \(g\), and the area is \(\int_a^b (f(x)-g(x))\,dx\). If \(D\) stays nonpositive, the area is \(\int_a^b (g(x)-f(x))\,dx\). A contact that does not change the sign does not change which expression applies on either side.
Why a Touch Does Not Require a Split
Suppose \(D(x)\geq 0\) on \([a,b]\), with \(D(c)=0\) at an interior point \(c\). Because \(D\) is nonnegative on both sides, the area expression remains \(D(x)\) across the whole interval. The additivity of definite integrals gives
So splitting at \(c\) is allowed, but it is not needed to correct the curve order: the same function remains above throughout. The same reasoning applies when \(D\leq 0\). By contrast, if \(D\) changes sign, the upper curve changes, and the area must be accounted for with the appropriate order on each side. A single contact point has zero width; its presence alone does not add or remove area.
Factoring the Difference to Detect a Touch
For polynomial functions, factoring their difference can make the behavior at a contact point clear. Suppose
where \(m\) is a positive integer, \(q\) is continuous near \(c\), and \(q(c)\ne 0\). Because \(q(c)\) is nonzero, continuity ensures that \(q\) keeps the same sign sufficiently close to \(c\). The sign behavior is then controlled by \((x-c)^m\): for even \(m\), this factor is positive on both sides of \(c\); for odd \(m\), its sign changes at \(c\). Thus an even power gives no order change locally, while an odd power gives a crossing.
This factor test applies to the stated polynomial factorization; it is not a claim that every differentiable difference must have such a factorization. It is a useful way to analyze many AP-level polynomial examples.
Worked Example: A Parabola Touches the Axis
Worked Example: A Parabola Touches the Axis
Find the area between \(y=x^2\) and \(y=0\) on \([-1,1]\), and decide whether the contact at \(x=0\) requires splitting the integral.
Set \(f(x)=x^2\) and \(g(x)=0\). Their difference is \(D(x)=x^2\). It is nonnegative on \([-1,1]\), and it equals zero only at \(x=0\). Therefore \(f\) is above \(g\) on both sides of the contact. The curves do not cross, so one integral gives the area:
At the contact, the slopes agree: \(f'(x)=2x\), so \(f'(0)=0\), and \(g'(x)=0\). To check the calculation, differentiating the antiderivative gives \(\frac{d}{dx}(x^3/3)=x^2\), the original integrand. The area is \(\frac23\) square units. Splitting at zero would give the same result, but is not necessary.
Equal Slopes Do Not Guarantee a Touch Without Crossing
At a differentiable contact, equal slopes are necessary for tangency, but they do not tell us whether the curves cross. The sign of the difference must still be checked. In particular, a difference can have derivative zero at a point and still change sign there.
Worked Example: Equal Slopes at a Crossing
Compare \(y=x^3\) and \(y=0\) on \([-1,1]\). Determine whether the curves touch without crossing or cross, then find their area.
The difference is \(D(x)=x^3\). At \(x=0\), the curves meet, and their slopes agree because \(\frac{d}{dx}(x^3)=3x^2\), which is zero at \(0\); the zero function also has slope zero. However, \(x^3<0\) for \(-1\leq x<0\), while \(x^3>0\) for \(0<x\leq 1\). The difference changes sign, so the curves cross.
The upper curve is \(y=0\) on \([-1,0]\) and \(y=x^3\) on \([0,1]\). Therefore,
Evaluating each part,
Thus \(A=\frac14+\frac14=\frac12\) square units. The check is that the derivative of \(-x^4/4\) is \(-x^3\) on the left, and the derivative of \(x^4/4\) is \(x^3\) on the right. Equal slopes at the contact did not prevent a crossing.
Worked Example: A Factored Difference with a Touch
Worked Example: A Factored Difference with a Touch
Find the area between \(y=(x-1)^2(x+2)\) and \(y=0\) on \([0,2]\). Identify what happens at their contact \(x=1\).
The difference is already factored:
On \([0,2]\), the factor \(x+2\) is positive, while \((x-1)^2\) is nonnegative and is zero only at \(x=1\). Hence \(D(x)\geq 0\) on the entire interval, and its sign does not change at the contact. The first curve remains above the axis, so use one integral. Expanding the integrand gives
Now integrate and evaluate:
As a check, differentiating the antiderivative gives \(x^3-3x+2\), which matches the expanded difference. The area is \(2\) square units. The squared factor explains why the order does not change at \(x=1\).
Common Mistake: Using Equal Slopes as the Only Test
A tempting but incorrect conclusion is: “The curves have equal slopes where they meet, so they must touch without crossing.” The cubic example shows why this fails. For \(y=x^3\) and \(y=0\), both slopes at zero are zero, but the difference \(x^3\) changes sign.
Another error is to integrate the signed difference and call the result area. For this crossing example, that incorrect calculation would be
Zero is the net signed integral, not the geometric area: the negative contribution on the left cancels the positive contribution on the right. The correction is to check the sign of \(D\) and use the upper-minus-lower difference on each side, giving total area \(\frac12\). In practice, use both slope information and a sign test; the sign test determines whether the curve order changes.
Choosing Bounds Around a Contact
Before evaluating an area integral, identify the interval whose area is being requested and examine the curve order throughout it. A contact at an endpoint does not, by itself, say what happens on the interior side. For an interior contact, compare the signs of \(f-g\) just to its left and right, or factor the difference when possible.
Solve \(f(x)=g(x)\) to locate where the graphs meet on the interval.
Set \(D(x)=f(x)-g(x)\) and determine its sign on either side of the contact.
The same sign on both sides means a touch without crossing; opposite signs mean a crossing.
Use upper minus lower. Split at a crossing where the upper curve changes; a touch does not require a split for curve order.
The essential distinction is between a point where the curves meet and a point where their order changes. A touch can be important for identifying the boundary of a region, but it does not automatically create a new upper-versus-lower expression. Let the sign of the difference, not the mere existence of an intersection, guide the setup.
Check Your Understanding
For each question, use the difference of the two functions to justify whether the contact changes curve order.
- For \(y=(x-2)^2\) and \(y=0\), what is the sign of the difference on either side of \(x=2\)? Is this a crossing?
- For \(y=(x-1)^3\) and \(y=0\), the slopes agree at \(x=1\). Do the curves cross there? Explain using signs.
- If \(f(x)-g(x)=(x+1)^2(x-3)\), what does the squared factor tell you about the order near \(x=-1\), provided the other factor is nonzero there?
- When the difference is nonnegative on an entire interval and is zero at one interior point, which curve is above, and must the area integral be split at that point?
- Why is \(\int_a^b(f(x)-g(x))\,dx\) not necessarily the geometric area between two curves?
