Answer
a) $f(x,y) = 3x + x^2y^2 + K$
b) $9$
Work Step by Step
$a)$
$F(x, y) = (3 + 2xy^2)i + (2x^2y)j$
$\frac{dP}{dy} = 4xy$
$\frac{dQ}{dx} = 4xy$
$\frac{dP}{dy} = \frac{dQ}{dx}$ , therefore $F(x,y)$ is conservative.
$f_x(x, y) = \int (3 + 2xy^2) dx$
$ = 3x + x^2y^2 + g(y)$
$fy(x, y) = \frac{d}{dy}(3x + x^2y^2 + g(y))$
$ = 2x^2y + g^1(y)$
$F = \nabla f$
$ = 3x + x^2y^2 + K$
$b)$
$\int_c Fdr = \int_c \nabla fdr$
$ = f(r(b)) - f(r(a))$
$ = f(4, \frac{1}{4}) - f(1, 1)$
$ = (3(4) + (4^2)(\frac{1}{4})^2) - (3(1) + (1)^2(1)^2)$
$ = (12 + 1) - (3 + 1)$
$ = 9$
