In 10.1

In[1]:= Simplify[Integrate[Integrate[If[{x^2/a^2+y^2/b^2<=1},1,0], {y,-Infinity,Infinity}], {x,-Infinity,Infinity}], {a>0,b>0}]

Out[1]= a b Pi

In[2]:= Simplify[Integrate[Integrate[If[x^2/a^2+y^2/b^2<=1,1,0], {y,-Infinity,Infinity}], {x,-Infinity,Infinity}], {a>0,b>0}]

Out[2]= a b Pi

In[3]:= Simplify[Integrate[Integrate[Boole[x^2/a^2+y^2/b^2<=1], {y,-Infinity,Infinity}], {x,-Infinity,Infinity}], {a>0,b>0}]

Out[3]= a b Pi

In[4]:= Simplify[Integrate[Integrate[Piecewise[{{1,x^2/a^2+y^2/b^2<=1}}, 0], {y,-Infinity,Infinity}], {x,-Infinity,Infinity}], {a>0,b>0}]

Out[4]= a b Pi

In[5]:= Simplify[Integrate[Integrate[Which[x^2/a^2+y^2/b^2<=1,1,True,0], {y,-Infinity,Infinity}], {x,-Infinity,Infinity}], {a>0,b>0}]

Out[5]= a b Pi

In[2]:= Integrate[
 If[x^2/a^2 + y^2/b^2 <= 1, 1, 0], {y, -Infinity, 
  Infinity}, {x, -Infinity, Infinity}]

Out[2]= \[Pi]/Sqrt[1/(a^2 b^2)]

It looks like you were trying to paste Traditional Math notation.

Please run Hold and InputForm on your integral:

putYourIntegralHere//Hold//InputForm

This will show the actual code that you are using, which you can then post here.