Suppose R is the shaded region in the figure. As an iterated integral in polar coordinates, B D f f(x, y) dA= [₁ fº R with limits of integration A = ? B = ? C = 0 D = 3sqrt(2) ← - ▶ 2 -3 [ f( cos(9), 7 sin(0)) r dr de -2 2 3
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- Evaluate | exp(-2²)dx, using an iterated integral and polar coordinates.Find polar coordinates (r,θ) where r < 0 and 0≤θ≤2π of(√3,−1)Let F = (5z + 5x) i + (7y + 4z+ 4 sin(y² )) j+ (5x + 4y + 7e*) .. (a) Find curl F. curl F = %3D (b) What does your answer to part (a) tell you about F. dr where C is the circle (x – 35)? + (y – 25)2 = 1 in the xy- %3D plane, oriented clockwise? fF. dr = 0 (c) If C is any closed curve, what can you say about F. dr? SF • dř = 0 (d) Now let C be the half circle (x – 35)2 + (y – 25)² = 1 in the xy-plane with y > 25, traversed from (36, 25) to (34, 25). Find F. dr by using your result from (c) and considering C plus the line segment connecting the endpoints of C. SF- dr =
- If the analytical function f(z) = u(x, y) + iv(x,y) is expressed in terms of the polar coordinates r and 0, show that 1 dv and or av 1 ди ar r 00 r ae Determine the analytical function f(z)=u+iv if = y'. Is u(x, y) harmonic? IIQuestion 10 Evaluate the double integral ff (2x+5y) dA by changing to polar coordinates where R is the region in the upper half plane bounded by the circles x2+y²=1 and x2+y²=9.Find the area of the region inside the lemniscate ²-6 cos 20 and outside the circle r=√√3. Sketch a graph of the region bounded by the graphs of the equations. Use: (1). Single integral in polar coordinate system (ii). Double integral in polar coordinate system.