Line Integrals - Part I — Question 9

Question 9 – Solution

Strategy. Express the line integral entirely in terms of the unknown endpoint coordinate $a$, then solve the resulting quadratic equation.

Step 1: Mass formula Use $$\mathbf r(t)=\langle at,at\rangle,\qquad 0\le t\le 1.$$ Then $\delta=2at$ and $ds=|\langle a,a\rangle|\,dt=a\sqrt 2\,dt$. Therefore $$M(a)=\int_0^1(2at)(a\sqrt 2)\,dt =2a^2\sqrt 2\left[\frac{t^2}{2}\right]_0^1 =a^2\sqrt 2.$$

Step 2: Recover $a$ The given mass implies $$a^2\sqrt 2=8\sqrt 2,\qquad a^2=8.$$ Because $a>0$, $$\boxed{a=2\sqrt 2}.$$

Step 3: Length and verification The diagonal length is $$L=\sqrt{a^2+a^2}=a\sqrt 2 =(2\sqrt 2)(\sqrt 2)=\boxed{4}.$$ Substitution into the mass formula gives $M=(2\sqrt 2)^2\sqrt 2=8\sqrt 2$, exactly the specified mass.