Double Integrals in Polar Coordinates1. A flat plate is in the shape of the region R in the first quadrant lying between the circles x2+ y2= 1and x2+ y2= 4. The density of the plate at point (x, y) is x + y kilograms per square meter (supposethe axes are marked in meters). Find the mass of the plate.R12x12y2. Find the area of the region R lying between the curves r = 2 + sin 3θ and r = 4 − cos 3θ. (You mayleave your answer as an iterated integral in polar coordinates.)- 55x- 55y3. In each part, rewrite the double integral as an iterated integral in polar coordinates. (Do not evaluate.)(a)ZZRp1 − x2− y2dA where R is the left half of the unit disk.-11-111(b)ZZRx2dA where R is the right half of the ring 4 ≤ x2+ y2≤ 9.-33-334. Rewrite the iterated integral in Cartesian coordinatesZ20Z√4−y2−√4−y2xy dx dy as an iterated integral inpolar coordinates. (Try to draw the region of integration.) You need not evaluate.5. Find the volume of the solid enclosed by the xy-plane and the paraboloid z = 9 − x2− y2. (You mayleave your answer as an iterated integral in polar coordinates.)xyz6. The region inside the curve r = 2 + sin 3θ and outside the curve r = 3 − sin 3θ consists of three pieces.Find the area of one of these pieces. (You may leave your answer as an iterated integral in polarcoordinates.)-44-442When doing integrals in polar coordinates, you often need to integrate trigonometric functions. Thedouble-angle formulas are very useful for this. (For instance, they are helpful for the integral in #2.)The double-angle formulas are easily derived from the facteit= cos t + i sin t (1)If θ is any angle, theneiθeiθ= e2iθ.Using (1) with t = θ on the left and t = 2θ on the right, this becomes(cos θ + i sin θ)(cos θ + i sin θ) = cos 2θ + i sin 2θcos2θ − sin2θ + 2i sin θ cos θ = cos 2θ + i sin 2θEquating the real parts of both sides, cos2θ − sin2θ = cos 2θ . Equating the imaginary parts,2 sin θ cos θ = sin 2θ .The formula cos 2θ = cos2θ − sin2θ also leads to useful identities for cos2θ and sin2θ:cos 2θ = cos2θ − sin2θ= cos2θ − (1 − cos2θ)= 2 cos2θ − 1cos2θ =12(1 + cos 2θ)cos 2θ = cos2θ − sin2θ= (1 − sin2θ) − sin2θ= 1 − 2 sin2θsin2θ =12(1 − cos 2θ)These two identities make it easy to integrate sin2θ and cos2θ.For the remaining problems, use polar coordinates or Cartesian coordinates, whichever seems easier.7. Find the volume of the “ice cream cone” bounded by the single cone z =px2+ y2and the paraboloidz = 3 −x24−y24.xyz38. A flat plate is in the shape of the region R defined by the inequalities x2+ y2≤ 4, 0 ≤ y ≤ 1, x ≤ 0.The density of the plate at the point (x, y) is −xy. Find the mass of the plate.9. Find the area of the region which lies inside the circle x2+(y−1)2= 1 but outside the circle x2+y2=
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