Find the average value of f over region D. f(x, y) = 6xy, D is the triangle with vertices (0, 0), (1, 0), and (1, 9)

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I have no idea how to find the limits for this. I've tried plotting it over x but cant figure it out.

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2 Answers

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Try the limits $0\le x\le 1$, $0\le y\le 9x$. Your exression is thus

$$\frac{\int_0^1 \int_0^{9x} 6xy\,dy\,dx}{\int_0^1 \int_0^{9x} \,dy\,dx}$$

Or, for the denominator, you could just use the area of the triangle.

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In general, if you have a triangle and you want inequalities that define that triangle, take the equation for the line between each pair of vertices and then let the last vertex determine whether your inequality should be greater than or less than for that side of the triangle. In this case it gives you $0 \leq x \leq 1$, $0 \leq y \leq 9x$, as indicated in another answer, which gives you limits for integration.

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