Find the volume of the solid under the graph of $f(x,y)$ over the region $R$ bounded by the graphs of the indicated equations. Sketch the region $R$; the solid does not have to be sketched.

$f(x,y)=4xy;R$ is the region bounded by the graphs of $y=\sqrt{1-x^2}$ and $y=0$ for $0\leq{x}\leq1$

Biology. In biology, there is an approximate rule, called the bioclimatic rule for temperate climates. This rule states that in spring and early summer, periodic phenomena such as the blossoming of flowers, the appearance of insects, and the ripening of fruit usually come about $4$ days later for each $500$ feet of altitude. Stated as a formula, the rule becomes

$$d=8h\qquad 0\leq{h}\leq4$$

where $d$ is the change in days and $h$ is the altitude (in thousands of feet). To test this rule, an experiment was set up to record the difference in blossoming times of the same type of apple tree at different altitudes. A summary of the results is given in the following table:

(A) Use the method of least squares to find a linear equation relating $h$ and $d$. Does the bioclimatic rule $d=8h$ appear to be approximately correct?

(B) How much longer will it take this type of apple tree to blossom at $3.5$ thousand feet than at sea level? [Use the linear equation found in part $(A)$.]

Evaluate each double integral. Select the order of integration carefully; each problem is easy to do one way and difficult the other.

$$\displaystyle\iint\limits_{R}xye^{x^2y}\ dA; R=\{(x,y)|0\leq{x}\leq1,1\leq{y}\leq2\}$$