Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2001 / Paper 2 / 8 / a

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- Solution a

Solution

The region of integration is

0 \leq y \leq 1

,

y \leq x \leq 2 - y

. It is the closed triangle with vertices

(0, 0)

,

(2, 0)

, and

(1, 1)

, bounded by

y = 0

,

x = y

, and

x = 2 - y

. Each horizontal slice runs from the rising left edge to the falling right edge.

Figure 1.
Triangular integration region and a horizontal integration slice
. Original diagram of the region $0 \leq y \leq 1$ , $y \leq x \leq 2 - y$ , with its three vertices, boundary equations, and an example slice at fixed $y$ . Details.

Integrate with respect to

x

I = \int_{0}^{1} [\frac{2}{3} x^{3} + y x]_{x = y}^{x = 2 - y} d y = \int_{0}^{1} [\frac{16}{3} - 6 y + 2 y^{2} - \frac{4}{3} y^{3}] d y = \frac{16}{3} - 3 + \frac{2}{3} - \frac{1}{3} = \frac{8}{3} .

The triangle has area one, but the nonconstant integrand makes the double integral a weighted area.

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