u Substitution in double integral

Question

$\mu_{Z}^{} = E(Z = \sqrt{(X^{2} + Y^{2})} = \int_{0}^{1}\int_{0}^{1}(x^{2}+y^{2})^{\frac{1}{2}}(4xy)dxdy$

Pulling out constant 4y

Step 1: $\mu_{Z}^{} = \int_{0}^{1}4y\int_{0}^{1}(x^{2}+y^{2})^{\frac{1}{2}}(x)dxdy$

applying u substitution

Step 2: $\mu_{Z} = \int_{0}^{1}{\frac{4y}{3}(x^{2}+y^{2})^{\frac{3}{2}}\mid_0^1}dy$

Step 3: $\mu_{Z} = {\frac{2}{15}}\left \{-2y^{5} + 2(y^{2}+1)^{}\frac{5}{2})\mid_0^1 \right \}$

= 0.9752

Now my question is, applying u substitution method how do I get from Step 1 to step 2. Any detail explanation of u substituion method will be appreciated.

Answer 1

Note that $$\frac{\partial}{\partial x}\bigg((x^2+y^2)^\frac32\bigg)=3x(x^2+y^2)^\frac12$$ So the integral $$\int_{0}^{1}x(x^{2}+y^{2})^{\frac{1}{2}}dx=\bigg[\frac13(x^2+y^2)^\frac32\bigg]_0^1=\frac13(1+y^2)^\frac32-\frac13y^3$$

Otherwise, one can use the substitution $u=x^2+y^2\Rightarrow du=2xdx$ to transform the integral $$\int_{0}^{1}x(x^{2}+y^{2})^{\frac{1}{2}}dx$$ into $$\int_{y^2}^{1+y^2}\frac12u^{\frac{1}{2}}du=\bigg[\frac13u^\frac32\bigg]_{y^2}^{1+y^2}=\frac13(1+y^2)^\frac32-\frac13y^3$$