$25\mathrm{.}1$ For the integral ${\displaystyle \underset{0}{\overset{2}{}}}{\displaystyle \underset{1}{\overset{5}{}}}\sqrt[\phantom{2}]{xy}dydx$ calculate the approximated values of integral $($a$)$ four$-$corners

method $($by hand, but use a calculator and a lot of digits$)$ using weight$-$matrix and formulas stated in the

class with $m=5($numbers of grid on the $x-$axis$)$ and $n=4($number of grid on the $y-$axis$)$

$($b$)$ Find the percentage error of these approximations, using the exact value. $($c$)$ Then, Modify the local

function myDoubleIntegral. $m$ to evaluate the integral using the four$-$corners method to evaluate

${\displaystyle \underset{0}{\overset{2}{}}}{\displaystyle \underset{1}{\overset{5}{}}}\sqrt[\phantom{2}]{xy}dydx$ when $(m,n)=(5,4),(m,n)=(10,8).$

$\%$ myDoublelntegral.m local function is to evaluate function values of $z=x^2+y^2$

$\%$ along the $x=0$:$5$:$10$ and $y=0$:$2$:$10.$

$\%$ The ultimate goal is to evaluate the double integral of $z$ over $[0,10]x[0,10].$

$dx=5$;

$dy=2$;

$x=0$:$dx$:$10$;

$y=0$:$dy$:$10$;

$z=$@$(x,y)(x^2+y^2)$;

$m=$ length $(x)$;

$n=$ length $(y)$;

$F=$zeros$(m,n)$;$\%m$ is the numbers of grid on the $x-$axis, $n$ is the numbers of grid on the $y-$axis.

for $j=1$:$n\%j$ is the index of column

for $i=1$:$m\%$ is the index of row

$F(i,j)=z(x(i),y(j))$;

end

end

$u=2*$ ones $(m,1)$;

$v=2*$ones $(n,1)$;

$u(1)=1$;

$u(m)=1$;

$v(1)=1$;

$v(n)=1$;

$A=u^{**}{v}^{\text{'}}$;

$,$ "all"$)$;

$I=T^{**}dx**d\frac{y}{4}$