Evaluate the line integral
SC F · dr, where C is given by the vector function r(t).
F(x, y, z) = sin x i + cos y j + xz k
r(t) = t^4i − t^3j + tk, 0 ≤ t ≤ 1
To evaluate the line integral
∫C F · dr
where C is given by the vector function r(t) and F(x, y, z) = sin(x)i + cos(y)j + xzk, we need to follow these steps:
1. Compute dr/dt:
Calculate the derivative of the vector function r(t) with respect to t.
dr/dt = d(t^4)i/dt - d(t^3)j/dt + dk/dt
= 4t^3i - 3t^2j + k
2. Substitute the components of r(t) and dr/dt into F:
F(x, y, z) = sin(x)i + cos(y)j + xzk
Substituting the components from r(t) and dr/dt:
F(r(t)) = sin(t^4)i + cos(-t^3)j + (t^4)(k)
3. Calculate the dot product F · dr:
F · dr = F(r(t)) · dr
= (sin(t^4)i + cos(-t^3)j + (t^4)(k)) · (4t^3i - 3t^2j + k)
Expand and simplify the dot product:
F · dr = sin(t^4)(4t^3) + cos(-t^3)(-3t^2) + (t^4)(1)
= 4t^7sin(t^4) + 3t^2cos(t^3) + t^4
4. Integrate the dot product over the given interval:
∫C F · dr = ∫[0,1] (4t^7sin(t^4) + 3t^2cos(t^3) + t^4) dt
Calculate this integral using standard integration techniques or computational methods.
Note: The intermediate steps for integrating the dot product may seem complex, but they involve basic vector operations such as calculating dot products and substituting vector components.
F·dr = sinx dx + cosy dy + xz dz
Now just crank it out, using the definitions of x,y,z:
∫4t^3 sin(t^4) - 3t^2 cos(t^3) + (t^4)(t) dt
= -cos(t^4) - sin(t^3) + 1/6 t^6 [0,1]
= -1 + 1/6
= -5/6