# Lake Volume

Calculus Level 2

The height profile of a valley basin can be described by the two-dimensional parabolic function$h(x, y) = \frac{x^2}{144\,\text{m}} + \frac{y^2}{324\,\text{m}}.$ Now the basin is filled by a rainstorm to a height of $$h_0 = 12\,\text{m}.$$ What is the volume of the resulting lake $$($$in $$\text{m}^3)$$ to the nearest integer?



Hint: Find the shape of the cross-sectional areas enclosed by the equipotential lines $$h(x, y) = z = \text{constant}$$. The volume then results from the integral $$\displaystyle V = \int_0^{h_0} A (z)\, dz$$ over the cross-sectional area $$A(z).$$

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