The following cell can be used to evaluate a double integral in the form:

(∫_a^b)((∫_(c[x])^(d[x]))f[x,y]dy)dx
i.e., when you integrate with respect to y first.

Clear[f,x,y,a,b,c,d];

f[x_,y_]=400-x^2-4*y^2;
a=-1;
b=1;
c[x_]=0;
d[x_]=Sqrt[1-x^2];

Integrate[
Integrate[f[x,y],{y,c[x],d[x]}]
,{x,a,b}]

N[%]

*****************************************

The following cell can be used to evaluate a double integral in the form:
(∫_a^b)((∫_(c[y])^(d[y]))f[x,y]dx)dy
i.e., when you integrate with respect to x first.

Clear[f,x,y,a,b,c,d];

f[x_,y_]=400-x^2-4*y^2;
a=0;
b=1;
c[y_]=-Sqrt[1-y^2];
d[y_]=Sqrt[1-y^2];

Integrate[
Integrate[f[x,y],{x,c[y],d[y]}]
,{y,a,b}]

N[%]

****************Plotting some boundaries*************************

ContourPlot[{y==0,y==x,x==0,x==2},{x,-1,4},{y,-1,4}]   (* 19 *)

ContourPlot[{y==x,y==2*x,x==1,x==5},{x,-1,6},{y,-1,15}]  (* 20 *)

ContourPlot[{x==Sqrt[y],x==y,y==1,y==4},{x,-5,5},{y,-5,5}] (* 21 *)

ContourPlot[{x==y,x==1,y==0,y==1},{x,-5,5},{y,-5,5}]  (* 34 *)

ContourPlot[{25-x^2-y^2==0},{x,-6,6},{y,-6,6}]  (* 42 *)

ContourPlot[{25-x^2-y^2==16},{x,-6,6},{y,-6,6}]  (* 43 *)