Example 2.4.1: Verifying the gradient theorem

$f(b)-f(a)=2^3\cdot 3-0=24$. The theorem predicts that both line integrals equal $24$.

with $d\ell =dx{a}_x+dy{a}_y$ in the plane.

On $C_1$: $y=0$, $dy=0$, $0\le x\le 2$, so the integrand $3x^2\cdot 0dx$ vanishes and ${\int }_{C_1}=0$. On $C_2$: $x=2$, $dx=0$, $0\le y\le 3$, so

Hence ${\int }_{C_1+C_2}\nabla f\cdot d\ell =0+24=24$.

The straight line through the origin and $(2,3)$ is $y=3x/2$, so $dy=\frac{3}{2}dx$ with $0\le x\le 2$. Then