5- y 4 R3 2- 1+ 4 5 6 x 18. Let R be the region in the first and second quadrants between the graphs of the polar curves f(0) = 3+3 cos 0 and g(0) = 4+2 cos 0, as shaded in the figure above. Which of the following integral expressions gives the area of R? (A) ₂ (8(0) -ƒ(0)) de (B) ſỗ (8(0) – ƒ (0)) dė (C) ½³ (8(0) — ƒ (e))² de (D) (((0))² - (f(0))²) də

Which integral represents the area of R? Choose 1 answer: •3π 4 A 6.5 (²1 + 2 cos² (0) + 2 cos*¹ (0)) d6 2 •3π 4 Ⓒ ³ (1 + cos² (0) + cos² (4) ) de B 4 3π √ ²³ ( ²7 + cos² (0) + cos² (0) do 3π 2 D Ⓒ / ² ( ² + 2 cos² (0) + 2 cos¹ (0)) de cos* (0)) de a

Choose the correct region of integration ²₁ √√4x² (x² + y²) dy dx. A D₁ D₂ D₁

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? II

Find the points (x, y) at which the polar curve r = 1+ sin(0), horizontal tangent line. Vertical Tangent Line: Horizontal Tangent Line: k|+ ㅠ 4 < 0 VI π A 4 has a vertical and

For a > 0, a lemniscate curve is the set of points P such that the product of the distances from P to (a, 0) and (−a, 0) is a2. Show that the equation of the lemniscate is (x2 + y2)2 = 2a2(x2 − y2) Then find the equation in polar coordinates. To obtain the simplest form of the equation, use the identity cos 2θ = cos2 θ − sin2 θ. Plot the lemniscate for a = 2 if you have a computer algebra system.

Let R = {(x, y) ER214 < x + y²< 9}. By inverting to polar coordinates: %3D dA = Vx? + y? R Select one: O None of these 3. 2 sin?(0)dodr. 3 27 r sin(0)dodr. 0. cos(0) sin(0)dodr. 0. sin (0)dodr.

(a). Express z = 3 + V3i in polar form. (b). Find the value of z if e- =1.

Q1/Prove that the polar expression for the function y = Vx? - (x2 + y?)3/2 is r= cos20

B B B B TBCN HUOY MO (B) 3.142 B (C) 8.566 y Unauthorized copying or reuse of any part of this page is illegal. B 91. The figure above shows the graphs of the polar curves r = 2 cos (30) and r = 2. What is the sum of the areas of the shaded regions? (A) 0.858 (D) 9.425 NAMUOY CELIAS ANT TIMO BIRA -X B T.30 THORD EKT MORA SEAL TRAS HONGY ITW 08 00 OT GOT BRA UOY TUMOROSE OT MO OD TOM 00 (E) 15.708 -35- 107 HVAR DOMEINE LOTTOMAT B BOY 31 CHEC A SEDAR OT MAUT ORAS 3502 e-re en REVEMA WEWA BUOY NOJESATASUMUM GA RUOY DAN SANOO AS HAL B GO ON TO THE NEXT PAGE