\(\vec F = xy\hat i + (x^2 + y^2)\hat j\)
\(\vec F.\vec {dr} = (xy\hat i + (x^2 + y^2)\hat j).(dx\hat i +dy\hat j)\)
\(= xydx + (x^2 + y^2)dy\)
Required
\(= \int \limits_c\vec F.\vec {dr}\), where \(c:y = x^2 - 4\)
\(= \int\limits_2^4x(x^2 -4)dx + (x^2 + (x^2 - 4)^2)(2xdx)\)
\(= \int\limits_2^4(x^3 -4x)dx + (2x^5 - 14x^3 + 32x)dx\)
\(= \int\limits_2^4 (2x^5 - 13x^3 + 28x)dx\)
\(= \frac 26 [x^6]_2^4 - \frac{13}4 [x^4]_2^4 + \frac{28}2[x^2]_2^4\)
\(= \frac 13 (4096 - 64) - \frac{13}4 (256 - 16 ) + 14(16 - 4)\)
\(= 1344 -780 + 168\)
\(= 1512 - 780\)
\(= 732\)
