compute the double integral $\\iint_{d} f(x,y) da$, where $f(x,y)=y^{2}\\sqrt{x}$ and $d$ is the set of…

Explanation:

Step1: Determine the limits of integration

First, find the intersection of (y = x^{2}) and (y=10 - x^{2}). Set (x^{2}=10 - x^{2}), then (2x^{2}=10), (x^{2} = 5). Since (x>0), (x=\sqrt{5}). The double - integral (\iint_{D}f(x,y)dA=\int_{0}^{\sqrt{5}}\int_{x^{2}}^{10 - x^{2}}y^{2}\sqrt{x}dydx).

Step2: Integrate with respect to (y)

Use the power rule for integration (\int y^{n}dy=\frac{y^{n + 1}}{n+1}+C) ((n\neq - 1)). (\int_{x^{2}}^{10 - x^{2}}y^{2}\sqrt{x}dy=\sqrt{x}\left[\frac{y^{3}}{3}\right]_{y = x^{2}}^{y = 10 - x^{2}}=\frac{\sqrt{x}}{3}\left[(10 - x^{2})^{3}-(x^{2})^{3}\right]). Expand ((10 - x^{2})^{3}=1000-300x^{2}+30x^{4}-x^{6}) using the formula ((a - b)^{3}=a^{3}-3a^{2}b + 3ab^{2}-b^{3}) ((a = 10), (b=x^{2})). So (\frac{\sqrt{x}}{3}\left[(10 - x^{2})^{3}-x^{6}\right]=\frac{\sqrt{x}}{3}(1000-300x^{2}+30x^{4}-2x^{6})).

Step3: Integrate with respect to (x)

Rewrite (\sqrt{x}=x^{\frac{1}{2}}). Then (\frac{1}{3}\int_{0}^{\sqrt{5}}(1000x^{\frac{1}{2}}-300x^{\frac{5}{2}}+30x^{\frac{9}{2}}-2x^{\frac{13}{2}})dx). Using the power rule (\int x^{n}dx=\frac{x^{n + 1}}{n + 1}+C) ((n\neq-1)): (\frac{1}{3}\left[1000\times\frac{2}{3}x^{\frac{3}{2}}-300\times\frac{2}{7}x^{\frac{7}{2}}+30\times\frac{2}{11}x^{\frac{11}{2}}-2\times\frac{2}{15}x^{\frac{15}{2}}\right]_{0}^{\sqrt{5}}). Substitute (x = \sqrt{5}) (and note that when (x = 0) the terms are (0)): [ \begin{align*} &\frac{1}{3}\left[\frac{2000}{3}(\sqrt{5})^{\frac{3}{2}}-\frac{600}{7}(\sqrt{5})^{\frac{7}{2}}+\frac{60}{11}(\sqrt{5})^{\frac{11}{2}}-\frac{4}{15}(\sqrt{5})^{\frac{15}{2}}\right]\ =&\frac{1}{3}\left[\frac{2000}{3}(5)^{\frac{3}{4}}-\frac{600}{7}(5)^{\frac{7}{4}}+\frac{60}{11}(5)^{\frac{11}{4}}-\frac{4}{15}(5)^{\frac{15}{4}}\right]\ \end{align*} ] [ \begin{align*} (\sqrt{5})^{\frac{3}{2}}&=5^{\frac{3}{4}},(\sqrt{5})^{\frac{7}{2}}=5^{\frac{7}{4}},(\sqrt{5})^{\frac{11}{2}}=5^{\frac{11}{4}},(\sqrt{5})^{\frac{15}{2}}=5^{\frac{15}{4}}\ \end{align*} ] [ \begin{align*} &\frac{1}{3}\left[\frac{2000}{3}\times5^{\frac{3}{4}}-\frac{600}{7}\times5^{\frac{7}{4}}+\frac{60}{11}\times5^{\frac{11}{4}}-\frac{4}{15}\times5^{\frac{15}{4}}\right]\ =&\frac{1}{3}\left[\frac{2000}{3}\times\sqrt[4]{125}-\frac{600}{7}\times\sqrt[4]{78125}+\frac{60}{11}\times\sqrt[4]{48828125}-\frac{4}{15}\times\sqrt[4]{476837158203125}\right]\ \end{align*} ] [ \begin{align*} &\frac{1}{3}\left[\frac{2000}{3}\times3.344- \frac{600}{7}\times16.701+\frac{60}{11}\times83.507-\frac{4}{15}\times417.537\right]\ =&\frac{1}{3}\left[2229.333-1431.514 + 455.498-111.343\right]\ =&\frac{1}{3}(1142)\ =&380.67 \end{align*} ]

Answer:

(380.67)