Measure the disk radius

The radius reaches from the axis to the curve

The teal region lies under the curve \(y = \sqrt{x}\) from \(x = 0\) to \(x = 9\), and it turns about the x-axis. The slider moves a slice, a thin vertical strip of the region.

The slice turns into a disk. The orange segment runs from the axis to the curve, so it reaches the edge of the disk. Its length is the radius.

At \(x = 4\) the curve is at height 2, so the radius is 2. At \(x = 9\) the curve is at height 3, so the radius is 3.

Move the slider. The radius changes with x, because the curve is higher where x is larger.

The radius is a distance from the axis

The strip is a thin piece of the region that stands across the dashed axis. The orange segment runs along the strip, from the axis to the boundary of the region, and its length is the radius.

For the x-axis, this segment is vertical, and its length is the height of the curve above the axis.

Press "Turn about the y-axis". The segment is now horizontal, and its length is the x-value of the curve, measured from the axis.

In both cases the radius changes as the slice moves, so you write it with the variable that runs along the axis.

Notation

1. About the x-axis, the boundary is a curve \(y = f(x)\), and the radius is its height above the axis.

   \[\cy{r(x)} = f(x)\]

2. About the y-axis, you write the boundary as \(x = g(y)\), and the radius is its x-value, measured from the axis.

   \[\cy{r(y)} = g(y)\]

3. If the curve is given as \(y = f(x)\), solve it for \(x\), which gives \(g(y)\).

4. The radius is the distance from the axis to the boundary, measured along the slice. A distance is not negative.

When it fails

The height \(f(x)\) can be negative. Press the button to move the region below the x-axis.

The curve now lies below the axis, so its height \(f(x)\) is negative. The radius is still the distance from the axis, which is the height without its sign, so here it is \(-f(x)\).

Example 1: the radius from a height

The region lies under \(y = \dfrac{8}{x + 1}\) from \(x = 0\) to \(x = 3\). It turns about the x-axis. Find the radius at \(x = 1\) and at \(x = 3\).

1. The axis is the x-axis, so the radius is the height of the curve.

   \[\cy{r(x)} = \frac{8}{x + 1}\]

2. At \(x = 1\), the orange segment is 4 units long.

   \[\frac{8}{1 + 1} = 4\]

3. At \(x = 3\), the curve is lower, so the segment is shorter.

   \[\frac{8}{3 + 1} = 2\]

Example 2: the radius from a width

The region lies between the y-axis, the curve \(y = 2x^2\) and the line \(y = 8\). It turns about the y-axis. Find the radius of the disk at height \(y\).

1. The axis is the y-axis, so the slices are horizontal. The radius is the horizontal distance from the axis to the curve.

2. A tempting wrong move: The slice at height \(y\) has radius \(y\).

   \[r = y\]

   At height 2, the curve passes through the point \((1, 2)\), so the slice reaches only 1 unit from the axis. The radius \(y\) would be 2, and the red disk reaches beyond the curve.

3. The radius is an x-value, so solve the curve for \(x\). The region lies to the right of the axis, so \(x\) is positive.

   \[y = 2x^2 \;\Rightarrow\; \cy{r(y)} = x = \sqrt{\tfrac{y}{2}}\]

4. At \(y = 2\) the radius is 1, and at \(y = 8\) it is 2. The orange segment ends on the curve.

   \[\sqrt{\tfrac{2}{2}} = 1, \quad \sqrt{\tfrac{8}{2}} = 2\]

Practice

The upper half of the circle \(x^2 + y^2 = 16\) turns about the x-axis and forms a ball. Find the radius of the disk at \(x = 2\).

Move the slider to \(x = 2\). The orange segment is the height of the circle at that \(x\). Solve the equation of the circle for \(y\).

Questions

The figure shows the region under the curve \(y = 6x - x^2\) from \(x = 0\) to \(x = 6\). It turns about the x-axis.

1. What is the radius of the disk at \(x = 2\)?

   • 8
   • 2
   • 6
   Answer

   8

2. At which value of x is the radius the greatest?

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   • 0
   Answer

   3

3. A region lies between the y-axis, the line \(y = 10\) and the curve \(y = x^2 + 1\). It turns about the y-axis. What is the radius of the disk at \(y = 5\)?

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   Answer

   2

Summary

You measure the radius as the distance from the axis to the boundary of the region, along the slice.

This works because the slice turns into a disk, and the distance from the axis to the edge of the disk is its radius.

Use the height of the curve about the x-axis, and the x-value of the curve about the y-axis.

This is one of four skills in topic 8.9 of Which & Why: AP Calculus.