Processing math: 100%
SOLUTION 4: If y=3+(4/5)x5/4 for 0≤x≤1, then dydx=(4/5)(5/4)x1/4=x1/4
so that
ARC=∫10√1+(dydx)2 dx
=∫10√1+(x1/4)2 dx
=∫10√1+x1/2 dx
=∫10√1+√x dx
(Now integrate using the method of "power" u-substitution. Let u=√1+√x ⟶
u2=1+√x ⟶ √x=u2−1 ⟶ x=(u2−1)2=u4−2u2+1 ⟶
dx=(4u3−4u)du .)
=∫x=1x=0√u2(4u3−4u) du
=∫x=1x=0u(4u3−4u) du
=∫x=1x=0(4u4−4u2) du
=((4/5)u5−(4/3)u3) |x=1x=0
=((4/5)(√1+√x)5−(4/3)(√1+√x)3) |10
=((4/5)(√1+√1)5−(4/3)(√1+√1)3)−((4/5)(√1+√0)5−(4/3)(√1+√0)3)
=((4/5)(√2)5−(4/3)(√2)3)−((4/5)(√1)5−(4/3)(√1)3)
=45(4√2)−43(2√2)−(1215−2015)
=(165−83)√2+815
=(4815−4015)√2+815
=815√2+815
=815(√2+1)
Click HERE to return to the list of problems.