C1 given by x^2+y^2 =1and C2 given by x^2+y^2 =9. Assume each is oriented counter-clockwise. Suppose that F = ⟨P, Q⟩ is a continuous vector field such that Py and Qx are continuous on the region D trapped between the two circles.
Suppose that integral(C1) F·dr=5 and integral(C2) F·dr=12. Calculate Integral IntegralD (Py −Qx)dA.
Suppose that integral(C1) F·dr=5 and integral(C2) F·dr=12. Calculate Integral IntegralD (Py −Qx)dA.
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Note that C₁ is the boundary of the unit disk and applying Green's Theorem you will integrate over the unit disk.
For C₂, it is the boundary of the disk with radius 3 centered at the origin. Applying Green's Theorem, you will integrate over the disk with radius 3 centered at the origin.
The region D is the annulus created by the two disks. Thus, since we given the values of the line integral along each of the curves, we know that
∬(D) (Py - Qx) dA = ∫(C₁) F•dr - ∫(C₂) F•dr = 5 - 12 = -7
Yin
For C₂, it is the boundary of the disk with radius 3 centered at the origin. Applying Green's Theorem, you will integrate over the disk with radius 3 centered at the origin.
The region D is the annulus created by the two disks. Thus, since we given the values of the line integral along each of the curves, we know that
∬(D) (Py - Qx) dA = ∫(C₁) F•dr - ∫(C₂) F•dr = 5 - 12 = -7
Yin