Use Green's theorem to evaluate the line integral along the given positively oriented curve, int_(C)7y^(3)dx-7x^(3)dy,quad" is the circle "x^(2)+y^(2)=4

Question

Use Green's theorem to evaluate the line integral along the given positively oriented curve,  int_(C)7y^(3)dx-7x^(3)dy,quad" is the circle "x^(2)+y^(2)=4

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Apply Green's Theorem:  Use Green's theorem to convert the line integral into a double integral over the region enclosed by the curve. Calculate Partial Derivatives:  Calculate the partial derivatives ∂Q/∂x and ∂P/∂y. Substitute into Double Integral:  Substitute the partial derivatives into the double integral. Convert to Polar Coordinates:  Recognize the region R as the disk x^2 + y^2 ≤ 4. Convert to polar coordinates where x = r cos(θ) and y = r sin(θ). Set up Limits for Integration:  Substitute polar coordinates into the integral and set up the limits for r and θ. Simplify Using Trigonometric Identity:  Simplify the integrand using the identity cos^2(θ) + sin^2(θ) = 1. Integrate with Respect to r:  Integrate with respect to r first. Integrate with Respect to θ:  Integrate the result with respect to θ.

Use Green's theorem to evaluate the line integral along the given positively oriented curve, $\int_{C}7y^{3}\,dx-7x^{3}\,dy,\quad \text{is the circle } x^{2}+y^{2}=4$

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Use Green's theorem to evaluate the line integral along the given positively oriented curve, ∫C7y3 dx−7x3 dy,is the circle x2+y2=4\int_{C}7y^{3}\,dx-7x^{3}\,dy,\quad \text{is the circle } x^{2}+y^{2}=4∫C​7y3dx−7x3dy,is the circle x2+y2=4

Use Green's theorem to evaluate the line integral along the given positively oriented curve, $\int_{C}7y^{3}\,dx-7x^{3}\,dy,\quad \text{is the circle } x^{2}+y^{2}=4$