Use the unsigned area, then subtract the rotating core

x=2t^2

Total marks: 9

Solution

(a)

Parametric area and integration by parts

A curve has parametric equations $x=2t^2$ , $y=\sin(-t)$ for $t\ge0$ . Find the exact area bounded by the curve, the $x$ -axis and the line $x=\frac{\pi^2}8$ .

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Hint

Find $t$ at the ends: $x=0$ and $x=\frac{\pi^2}{8}$ .
Is the curve above or below the axis? That decides the sign of $\int y\,dx$ .
Convert to $t$ : $dx=4t\,dt$ , then integrate $t\sin t$ by parts.

Answer · (a)

A=2\sqrt2-\pi\sqrt2/2

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A dip below the axis

Written in Cartesian form, the curve is y = −sin(√(x/2)). The strips run from the curve up to the x-axis, so each has a positive height.

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* = 0.

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

Sample x*: 0
Width Δx: 0.30843
Upper value: 0
Lower value: 0
Height: upper − lower: 0
Strip area: 0

0.30843 ×(0 − 0)≈ 0

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

Strip width

0.308

1.234 ÷ 4

Sum of strips

0.47462

Exact, 2√2 − π√2/2

0.60699

Error

−0.13236

21.81% too small

How the sum improves as n grows

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

Curve, below the x-axis
X-axis, the top of the region
Each has height (0 − y), a positive number
Exact area 2√2 − π√2/2 ≈ 0.6073

Key idea

The curve is below the axis, so $\int y\,dx$ is negative. Taking its negative gives an area of $2\sqrt2-\frac{\pi\sqrt2}{2}\approx0.607$ .

Ideas to try

Notice that every strip height is the positive number −y. This is what negating the integral does.
Use midpoints and n = 6. How close is the sum?

(b)

Volume of revolution

The curve $C$ has equation $y=x^2\sec x\tan^2x$ for $-\frac\pi2<x<\frac\pi2$ . The region $R$ is bounded by $C$ and the line $y=\frac{2\pi^2}3$ . Find the volume when $R$ is rotated through $2\pi$ radians about the $x$ -axis, giving your answer to 2 decimal places.

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Hint

Find where $C$ meets $y=\frac{2\pi^2}{3}$ . Try $x=\pm\frac\pi3$ .
Rotating about the $x$ -axis, washers have outer radius $\frac{2\pi^2}{3}$ and inner radius equal to $C$ .
The curve is even, so integrate over $[0,\frac\pi3]$ and double.

Answer · (b)

V=267.73

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A cylinder with a flared core removed

The flat top y = 2π²/3 is a cylinder's outside. The curve is nearly flat near the axis, then shoots up near ±π/3, so the hollow is narrow in the middle and wide at the ends.

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

Use the unsigned area, then subtract the rotating core

Solution

Parametric area and integration by parts

Limits in $t$

Take the sign into account

Convert to $t$

Integrate by parts

Simplify

A dip below the axis

Where does one strip’s area come from?

Volume of revolution

Find the limits

Cylinder minus the core

Use symmetry

Evaluate by calculator