Calculus 3 Green theorem

Question

Find the area of region enclosed by  using Green's Theorem.

Answer 1

-1/2 I[0,1] (x dy - y dx) =

-1/2 I[0,1] (t - t^6)(1 - 5t^4) - (t - t^5)(1 - 6t^5)dt =

-1/2 I[0,1]t - t^6 - 5t^5+5t^10 -(t - t^5 - 6t^6 + 6t^10)dt =

-1/2 I[0,1]5t^6-4t^5-t^10 dt =

-1/2 (5/7t^7 - 4/6t^6 - t^11/11) E[0,1] =

-1/2(5/7 - 4/6 - 1/11) = 5/231 = .021645

The negative occurs because the path is clockwise.

This is a well-known result for calculating areas in cases such as this.

Answer 2

Area A = ∫∫_Ddxdy = 1/2∫_Cxdy - ydx = 1/2∫₀¹[(t - t⁶)(1 - 5t⁴) - (t - t⁵)(1 - 6t⁵)]dt = 1/2∫₀¹(t - 5t⁵ - t⁶ + 5t¹⁰ - t + 6t⁶ +

t⁵ - 6t¹⁰) = 1/2∫₀¹(- 4t⁵ + 5t⁶ - t¹⁰)dt = 1/2(- 4/6 + 5/7 - 1/11) = - 0.021645.

A = 0.021645