Evaluate the line integral, where C is the given curve.
(Integral)C z dx + x dy + y dz,
C: x = t^4, y = t^5, z = t^4, 0 ≤ t ≤ 1
Just plug and chug
∫Cz dx + x dy + y dz
= ∫[0,1] (t^4)(4t^3) + (t^4)(5t^4) + (t^5)(4t^3) dt
= ∫[0,1] 4t^7 + 9t^8 dt
= 1/2 t^8 + t^9 [0,1]
= 1/2 + 1
= 3/2
To evaluate the line integral, we can follow these steps:
Step 1: Parameterize the given curve C.
Step 2: Compute the necessary derivatives.
Step 3: Substitute the parameterized values into the integrand.
Step 4: Evaluate the line integral over the given interval.
Let's go through each step in detail:
Step 1: Parameterize the given curve C.
The given curve C is defined by the equations:
x = t^4
y = t^5
z = t^4
Step 2: Compute the necessary derivatives.
To compute the derivatives, we differentiate each component with respect to the parameter t:
dx/dt = 4t^3
dy/dt = 5t^4
dz/dt = 4t^3
Step 3: Substitute the parameterized values into the integrand.
The integrand is given as:
z dx + x dy + y dz
Substituting the parameterized values, we have:
z = t^4
x = t^4
y = t^5
So, the integrand becomes:
t^4 * (4t^3) + t^4 * (5t^4) + t^5 * (4t^3)
Simplifying further:
4t^7 + 5t^8 + 4t^8
Step 4: Evaluate the line integral over the given interval.
We have the integrand as:
4t^7 + 5t^8 + 4t^8
To find the line integral, integrate the function with respect to t over the given interval [0, 1]:
∫(0 to 1) (4t^7 + 5t^8 + 4t^8) dt
Evaluating the integral will give us the final result.