Rotate the region between a bell curve and a line

y=2/[1+(x-2)^2]

Total marks: 9

Solution

(i)

Curve sketching

Sketch $y=\dfrac{2}{1+(x-2)^2}$ and $y=1$ on the same diagram. Label the stationary point and the intersection points.

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Hint

Where is $y$ largest? Look at when the denominator $1+(x-2)^2$ is smallest.
Where does the curve meet $y=1$ ? Solve $\frac{2}{1+(x-2)^2}=1$ .
What happens as $x\to\pm\infty$ , and where does the curve cross the $y$ -axis?

Bell curve area and washer geometryScroll across to see the full diagram.

Answer · (i)

(2,2)

Explore the idea

Add each feature an examiner looks for

Start with the bare curve and line. Switch on each feature to see where it goes and how you would find it.

Switch on a feature to add it to the sketch and see how it is found. A full-mark sketch shows all of them.

Bell curve y = 2/[1+(x−2)²]
Horizontal line y = 1
Line of symmetry x = 2

Ideas to try

Switch on “Meets y = 1” first. Could you predict both x-values from the symmetry alone?
Hover or tap along the curve to read off coordinates.

(ii)

Area between curves

Find the exact area of the region $R$ enclosed by the two curves and the lines $x=1$ and $x=3$ .

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Hint

Each thin vertical strip of $R$ has height (curve) − (line).
Integrate that height from $x=1$ to $x=3$ .
“Exact” means the answer stays in terms of $\pi$ , so look for an arctangent.

Answer · (ii)

=\pi-2

Explore the idea

An integral is the limit of a sum of strips

Approximate R with n vertical strips. Each has width 2/n and height curve − line. Add more strips and watch the total settle on the exact area.

Number of stripsn = 4

Try an example

Where does one strip’s area come from?

1 of 4

The curves meet at this sample, so the strip has zero height and contributes zero. Its red marks use the left edge at x* = 1.

Zoomed-in rectangle: width and height are scaled independently. The sample sets its upper and lower values across the whole strip.

Sample x*: 1
Width Δx: 0.5
Upper value: 1
Lower value: 1
Height: upper − lower: 0
Strip area: 0

0.5 ×(1 − 1)≈ 0

Add this contribution to those of the other 3 strips to get the sum below. Displayed values are rounded.

Strip width

0.500

2.000 ÷ 4

Sum of strips

1.10000

Exact, π − 2

1.14159

Error

−0.04159

3.64% too small

How the sum improves as n grows

The vertical scale fits n ≥ 3; earlier values may lie outside the view.

Rectangles of width 2/n. Each height is curve − line, measured at the left edge, midpoint or right edge
Exact area π − 2 ≈ 1.14159
Strip total for every n from 1 to 40. The dot marks your current n

Key idea

Strip totals converge to $\pi-2\approx1.14159$ , and the integral is exactly that limit. Midpoint strips get there fastest because each strip's overestimate on one side nearly cancels its underestimate on the other.

Ideas to try

Set n = 2 with the left-edge rule. Why do the strips miss so much of R?
Switch to midpoints and compare the error at n = 10 with the left-edge rule.
Drag n up to 40 and watch the error shrink towards zero.

(iii)

Volume of revolution

Find the volume generated when $R$ is rotated through $360^\circ$ about the $y$ -axis.

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Hint

Picture the solid. $R$ sits between $x=1$ and $x=3$ , so spinning it about the $y$ -axis leaves a hole.
For washers, slice horizontally. Each radius must be written as $x$ in terms of $y$ .
Outer radius from the right branch, inner radius from the left. Then integrate $\pi(R^2-r^2)$ over $y$ .

Answer · (iii)

=4\pi(\pi-2)\approx14.3

Explore the idea

Spin R about the y-axis, then slice the solid

Sweep R around the y-axis to build the solid. Then cut it into washers (horizontal slices) or shells (vertical tubes) and add up their volumes.

Open the interactive view to explore this geometry in 3D.

Takeaway

Connect the parts: area × distance travelled

By the symmetry from (i), the centre of $R$ is at $x=2$ . Each turn carries that centre around a circle of length $2\pi\times2$ . Multiply by the area from (ii) and you get the volume from (iii):

V=2\pi\cdot2\cdot(\pi-2)=4\pi(\pi-2)

This is Pappus’s centroid theorem. It is not on the syllabus, but it is a quick independent check of your answer.

Rotate the region between a bell curve and a line

Solution

Curve sketching

Find the stationary point

Find where the curves meet

Check the ends and the intercept

Draw it

Add each feature an examiner looks for

Area between curves

Set up upper minus lower

Spot the arctangent

Substitute the limits

Simplify and sense-check

An integral is the limit of a sum of strips

Where does one strip’s area come from?

Volume of revolution

Make x the subject

Area of one washer

Integrate over the heights of R

Evaluate

Spin R about the y-axis, then slice the solid

Connect the parts: area × distance travelled