Q 64. - Calculus

Question

Leila is designing a new summer range management unit for caribou in the Selkirk Mountains in the Idaho panhandle. The old unit was laid out as a rectangle, which

had nothing to do with the behavior of the caribou. The new one is supposed to resemble the actual area in which the caribou live. Leila has used a study which indicates that the density of caribou in this region in July is approximated by 

$d(x,y)=0.08x^2{y}^2-0.456x^{2}y-0.08x^2-$

$0.328x{y}^2+1.87xy+0.328x-0.061y^2+0.347y+0.061$.

Her proposed southern boundary for the management unit is a mountain ridge that roughly follows the curve $0.0195x^4$ while the northern border is a political boundary at $\frac{x}{4}+4$. The western boundary is a state line on which she places

the $y$-axis. Roughly how many caribou can be found in the management unit?

Step-by-Step Solution

Answer

There are $32$ caribous in management unit.

Step 1: Given Information

Density is given by $d(x,y)=0.08x^2{y}^2-0.456x^{2}y-0.08x^2-0.328x{y}^2+1.87xy$

$+0.328x-0.061y^2+0.347y+0.061$

Here, the region lies between $y=0.0195x^4\text{, the line }y=\frac{x}{4}+42$, and $y=0$

Step 2: Graph

The graph is

Step 3: Simplification

Evaluating the integral ${\iint }_{\Omega }d(x,y)dA$

$x$ varies from $0-4$, $y$ varies from $0.0195x^4\text{ to }\frac{x}{4}+4$

${\iint }_{\Omega }d(x,y)dA={\int }_0^4{\int }_{0.0195x^4}^{\frac{x}{4}+4}d(x,y)dydx$

${\int }_0^4{\int }_{0.0195x^4}^{\frac{x}{4}+4}d(x,y)dydx=32.000749$

${\int }_0^4{\int }_{0.0195x^4}^{\frac{x}{4}+4}d(x,y)dydx=32$

There are $32$ caribous in management unit.