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11th - a Z 2 π π Z 7 4 r dr dθ b Z π 2 Z 4 cos θ r dr...

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MATH 120 WEEK 11 (RECITATION QUESTIONS) 1. Find the area inside the larger loop and outside the smaller loop of the limacons r = 1 2 + cos θ . 2. Find the volume of the solid bounded by the cylinders x 2 + y 2 = r 2 and y 2 + z 2 = r 2 . 3. Use polar coordinates to combine the sum Z 1 1 2 Z x 1 - x 2 xy dy dx + Z 2 1 Z x 0 xy dy dx + Z 2 2 Z 4 - x 2 0 xy dy dx into one double integral. Then evaluate the double integral . 4. Evaluate the iterated integrals by converting to polar coordinates a ) Z 1 0 Z 1 - x 2 0 e x 2 + y 2 dy dx b ) Z a - a Z a 2 - y 2 0 ( x 2 + y 2 ) 3 2 dx dy 5. Sketch the region whose area is given by the integral and evaluate the integral
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Unformatted text preview: a ) Z 2 π π Z 7 4 r dr dθ b ) Z π 2 Z 4 cos θ r dr dθ 6. Use polar coodinates to ﬁnd the volume of the given solid; inside both the cylinder x 2 + y 2 = 4 and the ellipsoid 4 x 2 + 4 y 2 + z 2 = 64. 7. Find the area of the surface : the part of the plane z = 2 + 3 x + 4 y that lies above the rectangle [0 , 5] × [1 , 4]. 8. Find the area of the surface : the part of the sphere x 2 + y 2 + z 2 = 4 z that lies inside the paraboloid z = x 2 + y 2 . 1...
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