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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

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Find indefinite integrals using the substitutionright chevron icon

Full solution

Q. 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
  1. Apply Green's Theorem: Use Green's theorem to convert the line integral into a double integral over the region enclosed by the curve.
  2. Calculate Partial Derivatives: Calculate the partial derivatives ∂Q∂x\frac{\partial Q}{\partial x} and ∂P∂y\frac{\partial P}{\partial y}.
  3. Substitute into Double Integral: Substitute the partial derivatives into the double integral.
  4. Convert to Polar Coordinates: Recognize the region RR as the disk x2+y2≤4x^2 + y^2 \leq 4. Convert to polar coordinates where x=rcos⁡(θ)x = r \cos(\theta) and y=rsin⁡(θ)y = r \sin(\theta).
  5. Set up Limits for Integration: Substitute polar coordinates into the integral and set up the limits for rr and θ\theta.
  6. Simplify Using Trigonometric Identity: Simplify the integrand using the identity cos⁡2(θ)+sin⁡2(θ)=1\cos^2(\theta) + \sin^2(\theta) = 1.
  7. Integrate with Respect to rr: Integrate with respect to rr first.
  8. Integrate with Respect to θ\theta: Integrate the result with respect to θ\theta.

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