Is the domain for integral \( \int_{(1-x)^2}^\sqrt{1-x^2}f(x,y)dydx \) shown in the left picture or in the right?
LEFT
RIGHT answer± On the right one. Indeed,we have vertical slices starting at \( x=0 \), then for this value our slice is the just one point -- with \(y\)-value 1.
Match the regions with the corresponding double integrals
a,c,d,b,f,e.
a,d,e,c,f,b.
a,c,b,e,f,b. answer± We have two lines \y=x/2\) and \(y=x\) in the first integral, it is clearly "a", for the second we have line \(x=y \) so it is either "c" or "f", however, for "f" we don't have for any horizontal line intersection with domain given by interval of \( x\) from 0 to \(y \). Also, here we have the fith integral to match "f". The third integral is bounded above by \( y=x/2 \), so it is "d" for sure. And analogously, the forth is "b".
It is Hobbit time. The following regions resemble ancient runes of the Anglo
Saxons studied by JRR Tolkien. (A is "b", B is "d", C is "st", and D is "oe").
A,B,C,D
B,C,D,A
B,C,A,D
D,C,B,A answer± Set up all integrals. You can see that the first corresponds to B. Indeed we clearly have \( |x|\) and \( -|x| \) as the lower and upper bounds of the inner integral. The second is a slightly modification, and corresponds to C. Ok, the third looks a kind of weird. Skip it. What about the fourth one? It is square, and also relative to the first integral.
