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

Rui Baldwin 2021-06-13 Answered
Evaluate the line integral, where C is the given curve. y3 ds, C÷x=t3, y=t, 0? t? 3
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Expert Answer

opsadnojD
Answered 2021-06-14 Author has 95 answers

Step 1
f(x, y)=y3
r(t)=t3, t
r(t)=3t2, 1
ds=|r(t)|dt
ds=(3t2)2+12dt
ds=(9t4)+1dt
Integral y3ds
Integral =[0 to 3]f(r(t))ds
=[0 to 3]t3[(9t4)+1]dt
Let (9t4)+1=u
9×4t3dt+0=dut3dt=dt36
3(9t4)+1dt
udu36
(136)u32(32)+c
(154)u32+c
(154)[(9t4)+1]32+c
03t3(9t4)+1]dt=03(154)[(9t4)+1]32+c
(154)[(9×34)+1]32+c(154)[(9×04)+1]32c
(154)[730]32(154)
(154)[730321]
365.23

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