Evaluate the line integral, where C is the given curve. y3 ds, C: x = t3, y = t, 0 ? t ? 3

f(x,y)=y³

r(t)=<t³,t>

r'(t)=<3t²,1>

ds=|r'(t)|dt

ds=√[(3t²)² +1²]dt

ds=√[(9t⁴) +1]dt

integral y³ds

=integral_{[0 to 3]}f(r(t)).ds

=integral_{[0 to 3]} t³.√[(9t⁴) +1]dt

let (9t⁴) +1 =u

==>9*4t³dt+0=du==>t³dt=du/36

==>integralt³.√[(9t⁴) +1]dt

=>integral√u du/36

=>(1/36) u^3/2/(3/2) +c

=>(1/54) u^3/2+c

=>(1/54) [(9t⁴) +1]^3/2+c

integral_{[0 to 3]} t³.√[(9t⁴) +1]dt =_{[0 to 3]} (1/54) [(9t⁴) +1]^3/2+c

=>(1/54) [(9*3⁴) +1]^3/2+c -(1/54) [(9*0⁴) +1]^3/2-c

=>(1/54) [730]^3/2 -(1/54)

=>(1/54) [730^3/2 -1]

=>365.23