Answer
$f(x,y)=3x+x^2*y^2+K$
$9$
Work Step by Step
dP/dy=4xy
dQ/dx=4xy
This is a conservative vector field
Now I find the integral of vector I with respect to x
$\int(3+2xy^2)dx=3x+x^2y^2+f(y)$
which equals $f(x,y)$.
Now we need to find f(y)
We find it by taking the derivative in respect to y of f(x,y) and equal it to vector j.
d/dy(3x+x^2*y^2+f(y))=vector j (or dyf)
=2(x^2)y+f'(y)=2(x^2)*y
f'(y)=0 and taking the antiderivative of 0 is a constant
f(y)=C or K
Now plug in K to f(y) giving us
f(x,y)=3x+(x^2)(y^2)+K
Now we do FTC by plugging in f(b)-f(a)
b=(4,1/4) a=(1,1)
Plug in f(4,1/4)-f(1,1) in f(x,y)
3(4)+16*1/16-(3+1)=12+1-4=9
