MHB How to prove this corollary in Line Integral using Riemann integral

[WMDhamnekar]

. Let C be a smooth curve with arc length L, and suppose that f(x, y) = P(x, y)i +Q(x, y)j is a vector field such that $|| f|(x,y) || \leq M $ for all (x,y) on C. Show that $\left\vert\displaystyle\int_C f \cdot dr \right\vert \leq ML $
Hint: Recall that $\left\vert\displaystyle\int_a^b g(x) dx\right\vert \leq \displaystyle\int_a^b |g(x) | dx $ for Riemann Integrals.

Now, how to prove this corollary using this hint?

 

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Assuming $C$ is parametrized by $r(t)$, $a\le t \le b$, write $$\int_C f\cdot dr = \int_a^b f(r(t))\cdot r'(t)\, dt$$ By the hint, $\int_C f\cdot dr$ is bounded by $\int_a^b |f(r(t))\cdot r'(t)|\, dt$. Since $|f(r(t))\cdot r'(t)| \le |f(r(t))| |r'(t)| \le M |r'(t)|$ for all $t$, we deduce $\left|\int_C f\cdot dr\right| \le M \int_a^b |r'(t)|\, dt = ML$.