Question:

a) Evaluate the integral \(\int\limits_{0}^{1}{\int\limits_{0}^{1-{{x}^{2}}}{\left( {{x}^{2}}+{{y}^{2}} \right)dydx}}\)

b) Let \(w=w\left( u,v \right)\), \(u=\frac{x}{y}\) and \(v={{x}^{2}}+{{y}^{2}}\).

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Express the partial derivatives \(\frac{\partial w}{\partial x}\) and \(\frac{\partial w}{\partial y}\) in terms of the partial derivatives of w with respect to u and v.

c) Solve the initial value problem: where y(/2) = -2.

d) Solve the initial value problem: , with and

e) You have taken 12 measurements for an experiment in lab, however when you get hoMeand start to write the lab report you only have the following values recorded in your notebook:

17, 26, 31, 19, 25, 20, 19, 29, 11, 27

In your lab notebook, you have noted that the mean of the data (including the two missing data points) is 29 and also you remember that the two missing data points were the same value. What is the value of the two missing data points?