To solve the given line integral using Stokes' Theorem, let's break it down: ### Stokes' Theorem Stokes' Theorem relates a surface integral of the curl of a vector field over a surface to the line integral of the vector field around the boundary of the surface. The theorem states: \[ \int_C \mathbf{F} \cdot d\mathbf{l} = \iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S} \] Where: - \( C \) is the closed curve (boundary of the surface \( S \)), - \( \mathbf{F} \) is the vector field, - \( \nabla \times \mathbf{F} \) is the curl of \( \mathbf{F} \), - \( d\mathbf{S} \) is the vector surface element. ### Given Problem The vector field is: \[ \mathbf{F} = y \hat{i} + xz^3 \hat{j} - zy^3 \hat{k} \] And the curve \( C \) is a circle defined by: \[ x^2 + y^2 = 4 \quad \text{in the plane} \quad z = -3 \] ### Step 1: Compute the Curl of \( \mathbf{F} \) The curl of a vector field \( \mathbf{F} = F_1 \hat{i} + F_2 \hat{j} + F_3 \hat{k} \) is given by: \[ \nabla \times \mathbf{F} = \left( \frac{\partial F_3}{\partial y} - \frac{\partial F_2}{\partial z} \right) \hat{i} - \left( \frac{\partial F_3}{\partial x} - \frac{\partial F_1}{\partial z} \right) \hat{j} + \left( \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y} \right) \hat{k} \] Let's compute the components of the curl: - \( F_1 = y \), \( F_2 = xz^3 \), and \( F_3 =