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Areas Between Curves
Calculus 2 · Chapter 4: Applications of IntegrationSection 4.1 · lesson 16 of 5411 min · free
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In this lesson
When the x-axis is replaced by a real lower boundary.
- Set up area as the integral of top minus bottom
- Find limits by solving for the intersection points
- Split the region wherever the curves cross
- Choose dx or dy slicing to avoid splitting entirely
Transcript
Everything said in the video, section by section (1,691 words). Read along, or search it.
MAT-265 - Calculus 2 · Section 4.1
Areas Between Curves
A civil engineer planning a park on a parcel bounded by two winding creeks needs the exact acreage of the strip of land between them. A financial analyst comparing two investment funds wants the cumulative gap in their growth — the area trapped between two performance curves over a decade. An environmental scientist measuring water volume in a reservoir cross-section computes the area between the water surface and the sloping bed. In every one of those cases, the quantity of interest is not the area under a single curve. It's the area between two curves. So the single definite integral that measured area under one function now becomes a difference of two functions, integrated across the interval where they bound a region. And here's the reassuring part. There's almost no new machinery.
The Idea
The x-Axis Was Just a Special Curve
Here's the key insight, and it reframes everything you already know. In introductory integration, the definite integral of f measures the signed area between a curve and the x-axis. But the x-axis is just a special curve. It's y equals zero. Replace it with a genuine lower boundary, and area under a curve becomes area between two curves. So suppose two continuous functions satisfy f of x at least g of x on an interval. The graph of f lies above the graph of g. Slice the region between them into thin vertical rectangles. A representative slice at position x has height equal to the vertical distance between the curves — f of x minus g of x — and an infinitesimal width d x. Its area is that height times that width. Summing all such slices — that is, integrating — gives the total area. So the formula is the integral of top minus bottom. And because the integrand is the height of a slice, it must be upper curve minus lower curve. Get that order right and the result is always nonnegative. Now here's a crucial advantage worth appreciating. This formula works even when part or all of the region lies below the x-axis. The subtraction measures the gap between the curves directly. So no separate handling of negative y-values is needed. Which removes a very common source of error — students computing two separate areas under curves and then trying to combine them with the right signs.
Figure 4.1.1
One Representative Slice Tells You Everything
Let's look at a concrete region. Bounded above by y equals six minus x squared — a downward parabola. And below by y equals x — a straight line. The shaded band between them is what we want. And that red sliver is the representative slice. Width d x, height top minus bottom, which is six minus x squared, minus x. The curves cross at minus three comma minus three, and at two comma two. Those crossings set the limits of integration. Now notice something. Part of this region lies below the x-axis, down in the third quadrant. That doesn't matter at all. The top-minus-bottom difference handles it automatically. So set up the integral from minus three to two, of six minus x, minus x squared. Integrate term by term: six x, minus x squared over two, minus x cubed over three. Evaluate at x equals two and you get twenty-two thirds. At x equals minus three you get minus twenty-seven halves. Subtract, and the area is one hundred twenty-five over six. About twenty point eight three square units.
Finding the Limits
Set the Functions Equal - and Watch for Crossings
The limits of integration are rarely handed to you. You have to find where the curves meet. Set the two functions equal, and solve. For that last example, six minus x squared equals x. Rearrange to x squared plus x minus six equals zero. Factor as x plus three, times x minus two. Giving x equals minus three and x equals two. Those are your limits. But now a subtler situation, and this is where people lose points. What if the two curves cross inside the interval? So that the top and bottom roles swap partway through? Over one part of the interval f is above g. Over another part, g is above f. A single integral of f minus g would then subtract a negative height somewhere, and undercount the area. The remedy is to split the integral at each crossing point, and use the correct top-minus-bottom ordering on each piece. Let's see it. Find the total area between sine x and cosine x, over zero to pi over two. First find where they cross inside the interval. Sine equals cosine at x equals pi over four. Now test each subinterval. On zero to pi over four, cosine is on top — check at x equals zero: cosine is one, sine is zero. On pi over four to pi over two, sine is on top. So split accordingly. Two integrals, each with its own correct ordering. The first evaluates to root two minus one. The second also evaluates to root two minus one. Add them: two root two minus two. About zero point eight three. Now — what would have happened without splitting? A single integral of sine minus cosine over the whole interval gives exactly zero. The two equal pieces cancel perfectly. A serious undercount, and nothing in the arithmetic warns you.
Slicing Direction
Sometimes dy Replaces Two Integrals With One
Vertical slices, width d x, are the default. But some regions are far easier with horizontal slices, width d y. And the choice hinges on which orientation lets a single formula describe the region without splitting. Slice vertically — integrate in x — when the region has a single upper curve and a single lower curve across the whole interval. That means every vertical line crosses the region entering through one curve and exiting through another. Slice horizontally — integrate in y — when the left and right boundaries are single curves, but the top or bottom boundary changes. Which often happens when the bounding curves are naturally written as x in terms of y. The formula becomes the integral of right minus left, d y. Conceptually it's identical. A slice is still a thin rectangle whose dimensions are read off the region. The roles of the variables are just exchanged. Let's see a case where it matters. Find the area enclosed by the parabola x equals y squared, and the line x equals y plus two. If you solve x equals y squared for y, you get two branches — plus and minus root x. So vertical slicing would require splitting the region. Horizontal slicing avoids that entirely, because both boundaries are already written as x in terms of y. Set the right sides equal: y squared equals y plus two. So y squared minus y minus two equals zero, factoring as y minus two, times y plus one. So y is minus one and two. For y between those values, the line lies to the right of the parabola. Check at y equals zero: the line gives x equals two, the parabola gives x equals zero. Yes. Apply right minus left, integrate, and the area is nine halves. Four point five. One clean integral, where d x slicing would have needed two.
Reference
Choosing the Slicing Direction
Let's summarize the choice. Vertical slicing, integrating in x, with integrand top minus bottom. Best when you have one upper and one lower curve, both as functions of x. Horizontal slicing, integrating in y, with integrand right minus left. Best when boundaries are given as functions of y, or the curves open sideways. Recognizing which variable to integrate in is often the difference between a one-line solution and a tangle of separate cases. So it's worth a moment of inspection before committing to a setup. Here's something to think about. In that last example, the region could in principle be computed with vertical slices too. But it would require splitting into two integrals at x equals one. Because for x between zero and one, both the top and the bottom of the region are branches of the parabola. While for x between one and four, the top is the parabola and the bottom is the line. Two different descriptions. Two integrals. So before reaching for a formula, what should you ask about a region to decide whether d x or d y slicing will need fewer pieces? What feature of the boundary curves is the clearest signal?
Avoiding Errors
Two Related Mistakes Dominate
Two related mistakes dominate this topic. The first is integrating bottom minus top. That produces a negative number. And here's the useful thing — a negative area is impossible. So if an area computation yields a negative result, the curve labeling was reversed. The fix is simply to swap the order, or take the absolute value. That error announces itself, which makes it the safer of the two. The second is failing to split when curves cross inside the interval. As in the sine and cosine example, the cancellation silently undercounts the area and gives a confidently wrong answer. There's no negative sign to warn you. Nothing looks off. The safeguard for both is the same, and it takes thirty seconds. Sketch the region first. Identify which curve is on top — or on the right — over each subinterval. And only then set up the integral. A quick test value in each subinterval confirms the ordering before any integration begins.
Key Takeaways
What to Carry Forward
Five things to carry forward. Area between curves is the integral of top minus bottom in x. Or equivalently, right minus left in y. The integrand is the height or width of a representative slice. Find the limits by solving f of x equals g of x. Those intersection points set the bounds, and they also reveal any crossings. Split the region at every interior crossing, and use the correct ordering on each piece. Otherwise the canceling pieces undercount the area. Choose d y slicing when the boundaries are functions of y, or when curves open sideways. It often replaces two d x integrals with one. And a negative area signals reversed ordering. Sketch first, and test a point in each subinterval to fix the top and bottom labels. Next up is section four point two — Volumes by the Disk and Washer Methods, where this same slicing idea builds solids instead of regions.
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