AP Calculus AB and BC glossary

Radius Function

Also called: Outer and inner radius, Radius of a washer

Every radius function measures the distance from the axis of revolution to one boundary of the region. In a disk or washer integral that distance is the quantity that gets squared; in a shell integral the same distance is multiplied by the height instead. A washer needs two at once, an outer R and an inner r.

R(x)=∣k−ffar(x)∣,r(x)=∣k−fnear(x)∣R(x)=\bigl|k-f_{\text{far}}(x)\bigr|, \quad r(x)=\bigl|k-f_{\text{near}}(x)\bigr|

Read the radius off the picture rather than off the function. Revolving the region under y=f(x)y=f(x) about the xx axis gives radius f(x)f(x), and the slices are solid disks because the axis borders the region. Revolving that same region about y=−1y=-1 gives an outer radius of f(x)+1f(x)+1 and, because the axis no longer touches the region, an inner radius of 11, so every slice is a washer. Keeping only the far radius and integrating πR2\pi R^2 builds a solid with its hole filled back in.

V=π∫ab[ R(x)2−r(x)2 ]dxV = \pi\int_a^b \left[\,R(x)^2 - r(x)^2\,\right] dx

The two radii travel together. RR runs from the axis to the far boundary of the region and rr from the axis to the near boundary, both measured from the same line, so moving the axis moves both at once. Revolving that region about y=3y=3 with the curve below the line gives R=3R=3, the distance out to the xx axis, and r=3−f(x)r=3-f(x), the distance in to the curve.

The mistake

Writing (R−r)2(R-r)^2 where the integrand needs R2−r2R^2-r^2. A washer is an outer disk with an inner disk cut out, so the two areas subtract only after each radius has been squared. Squaring the gap between the radii measures a completely different circle, and the two expressions agree only when r=0r=0 or r=Rr=R.

Appears in: Unit 8: Applications of Integration