4. (a) Evaluate \( \oint_{C}\left(3 y-e^{\sin x}\right) d x+\left(7 x+\sqrt{y^{4}+1}\right) d y \), where the region \( D \) enclosed by \( C \) is given by \( D=\{(r, \theta): 0 \leq r \leq 3,0 \leq \theta \leq 2 \pi\} \)

To evaluate the line integral \( \oint_{C}\left(3 y-e^{\sin x}\right) d x+\left(7 x+\sqrt{y^{4}+1}\right) d y \) over the region \( D \), we can apply Green's Theorem, which relates a line integral around a simple closed curve to a double integral over the region it encloses. Using Green’s Theorem: \[ \oint_{C} (P \, dx + Q \, dy) = \iint_{D} \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) dA \] where \( P = 3y - e^{\sin x} \) and \( Q = 7x + \sqrt{y^4 + 1} \). 1. Compute \( \frac{\partial Q}{\partial x} \): \[ \frac{\partial Q}{\partial x} = 7 \] 2. Compute \( \frac{\partial P}{\partial y} \): \[ \frac{\partial P}{\partial y} = 3 \] 3. Apply these to Green's Theorem: \[ \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} = 7 - 3 = 4 \] 4. Next, we find the area of region \( D \), which is a disk of radius 3: \[ \text{Area} = \pi r^2 = \pi (3^2) = 9\pi \] 5. Finally, we compute the double integral: \[ \iint_{D} 4 \, dA = 4 \times \text{Area} = 4 \times 9\pi = 36\pi \] Thus, the value of the integral is: \[ \oint_{C}\left(3 y-e^{\sin x}\right) d x+\left(7 x+\sqrt{y^{4}+1}\right) d y = 36\pi \]