Shells for a Line-Parabola Region

From Two Curves to a Shell Integral

A region bounded by a line and a parabola has a curved boundary and a straight boundary. To find the volume formed by rotating such a region about the \(y\)-axis, we first locate where the graphs meet, then determine which graph lies above the other between those points.

Using vertical slices produces cylindrical shells. Each shell’s radius is its distance from the \(y\)-axis, and its height is the vertical distance between the two curves. Once these quantities and the intersection bounds are known, the volume follows from a definite integral.

Finding the Bounds and the Shell Height

The intersection points occur where the equations of the line and parabola have the same \(y\)-value. Equating them gives an equation in \(x\); its solutions are the \(x\)-coordinates of the intersections. For a bounded region formed by the two graphs, these intersection coordinates typically provide the integration bounds.

To decide which graph is on top between the bounds, subtract one function from the other. The difference is the shell height only when its sign and ordering have been identified. If the line is above the parabola, for example, the height is line minus parabola. A test value strictly between the intersection coordinates can confirm the sign.

For vertical shells about the \(y\)-axis, the shell radius is \(x\) when the region lies at \(x\geq 0\). The shell-method formula then multiplies circumference, \(2\pi x\), by height and thickness, \(h(x)\,dx\). If a region crosses the axis, extra care is needed because slices on opposite sides can generate overlapping shells; the simple integral over signed \(x\) does not automatically give the volume.

Worked Examples

Why the Curve Ordering Is Reliable

The shell height must be nonnegative, so choosing the upper curve matters. In these examples, the difference between the line and parabola is a quadratic with the two intersection coordinates as roots. Its factorization makes the sign on the interval particularly easy to establish.

To verify the result, take \(a<x<b\). Then \(x-a>0\), while \(x-b<0\), so \((x-a)(x-b)<0\). Multiplication by \(-k\), with \(k>0\), gives \(D(x)>0\). At \(x=a\) and \(x=b\), the difference is zero, so the curves meet at both endpoints. This proves the ordering across the entire interval, rather than only at one test point.

More generally, if the curves have no other intersections inside the interval, their difference cannot change sign there: a change from positive to negative would require the difference to equal zero somewhere in between. For a quadratic difference whose only roots are the two endpoints, that cannot happen. Factoring the difference or testing one interior value are both useful ways to identify the shell height.

Common Mistake: Reversing the Height

A frequent error is to subtract the upper curve from the lower curve. For the second example, that mistake would use \(x^2-(5x-6)=x^2-5x+6\). At \(x=\frac{5}{2}\), it gives \(-\frac{1}{4}\), a negative shell height. The negative result signals that the subtraction order is reversed.

The correction is to use upper minus lower: \(5x-6-x^2\). At the same test value this is \(\frac{1}{4}>0\). With the correct order, the integral gives \(\frac{5\pi}{6}\), rather than a negative volume. A volume cannot be negative; a negative setup should prompt a check of the curve ordering, bounds, and radius.

A Reliable Setup Routine

For a bounded region between a line and a parabola rotated about the \(y\)-axis, keep the steps in this order:

The central task is not the integration itself; it is translating the geometry into the correct bounds, radius, and height. Once those are set correctly, the shell integral is a direct calculation.