Question

Let \( C \) be the circle \( (x - 1)^2 + y^2 = 1 \), oriented counterclockwise. Then the value of the line integral \[ \int_C \left( \frac{4}{3} x y^3 \, dx + x^4 \, dy \right) \]

• A. \( 6\pi \)
• B. \( 8\pi \)
• C. \( 12\pi \)
• D. \( 14\pi \)

Hint:

When solving line integrals over a closed curve like a circle, parameterize the curve and simplify the integral as much as possible before solving.

Answer 1

The Correct Option is B

Step 1: Parameterize the circle.
The equation of the circle is \( (x - 1)^2 + y^2 = 1 \), which has a center at \( (1, 0) \) and radius 1. We can parameterize the curve as: \[ x = 1 + \cos t, \quad y = \sin t, \quad t \in [0, 2\pi]. \]
Step 2: Substitute into the integral.
The line integral is: \[ \int_0^{2\pi} \left( \frac{4}{3} (1 + \cos t) \sin^3 t (-\sin t) + (1 + \cos t)^4 \cos t \right) \, dt. \]
Step 3: Simplify and compute.
After simplifying the integral, we calculate the integral and find that the value is \( 8\pi \).

Step 4: Conclusion.
Thus, the correct answer is \( \boxed{(B)} \).