Question

Let f ( x , y )=5 x and let C be the segment of the parabola y = 2 x squared joining O( 0 , 0 ) and P( 1 , 2 ). a. Find a parameterization of C in the direction from O to P. Evaluate Integral from Upper C to f ds. b. Find a parameterization of C in the direction from P to O. Evaluate Integral from Upper C to f ds.

Answer 1

a. We can parameterize $C$ by

$x=t$

$y=2t^2$

with $0\le t\le1$. Then

$\displaystyle\int_C5x\,\mathrm dS=\int_0^15t\sqrt{1+16t^2}\,\mathrm dt=\dfrac5{48}(17^{3/2}-1)$

b. We can parameterize the opposite direction by instead setting

$x=1-t$

$y=2(1-t)^2$

with $0\le t\le 1$. Then

$\displaystyle\int_C5x\,\mathrm dS=\int_0^15(1-t)\sqrt{1+16(1-t)^2}\,\mathrm dt=-\int_1^05u\sqrt{1+16u^2}\,\mathrm dt=\int_0^15u\sqrt{1+16u^2}\,\mathrm du$

which gives the same value as in part (a).