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# MultivariateCalculus MATH200

Drexel

GPA 3.63

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This 2 page Class Notes was uploaded by Moshe Swift III on Wednesday September 23, 2015. The Class Notes belongs to MATH200 at Drexel University taught by RobertBoyer in Fall. Since its upload, it has received 28 views. For similar materials see /class/212285/math200-drexel-university in Mathematics (M) at Drexel University.

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Date Created: 09/23/15

WINTER 2004 DOUBLE INTEGRALS IN POLAR COORDINATES 7 REVIEW MATH 200 H F 9quot th 9quot 93 T1 F3 Evaluate ydA where R is the region in the rst quadrant bounded by the circles 952 y2 4 and R x2 y2 25 Comment Look at 0 250 sin9rdrd9 Evaluate Mac 32 CIA where D is the region bounded by the cardioid r 1 cos 9 D Comment Look at 0 OHCOSQ r r dr d9 Evaluate di where D is the region in the rst quadrant that lies between the circles 952 y2 4 and D x2 y2 295 Comment Note that x2 y2 295 is really the circle as 7 12 y2 1 with center at 1 0 and radius 1 ln polar coordinates this circle has the form r 2cos 9 Note at 9 0 r 2 while at 9 7r2 r 0 7r2 Look at di rdrd9 D 0 2cos9 Find the area of the region enclosed by the cardioid r 1 7 sin 9 27r 17sin9 Comment Look at dA rdrd9 D 0 0 Find the area of the region inside the circle r 3cos9 and outside the cardioid r 1 cos 9 Comment We need to nd the two intersection points 3cos9 1 cos9 so 2 cos9 1 or 9 i7r3 The s9 7r 3 3 co desired integral is then r dr d9 77r3 1cos9 Find the volume of the solid under the cone 2 M952 y2 and above the ring 4 S x2 y2 S 25 27r 5 Comment x2 y2 dA r r drd9 D 0 2 Find the volume ofthe solid that is bounded above by the the sphere x2y222 1 and below by 2 x x2 3 Comment In polar coordinates the cone is 2 r and the sphere is r2 22 1 The volume is given by 27r 195 V 17 r2 7 r rdrd9 0 0 Find the volume of the solid bounded by the paraboloids 2 3952 32 and 2 4 7 x2 32 Comment In polar coordinates the paraboloids are 2 3r2 and 2 4 7 r2 They intersect at 3r2 4 7 r2 7r or r 1 We get 4 7 r2 7 3r2 rdr d9 0 0 Find the volume of the solid that inside both the cylinder 952 y2 4 and the ellipsoid 4952 4y2 22 64 Comment In polar coordinates the cylinder is r 2 and the ellipsoid is 4r2 22 64 The integral is 7 27r 6 r dr d9 0 74647 4 WINTER 2004 TRIPLE INTEGRALS IN SPHERICAL COORDINATES 7 REVIEW MATH 200 1 Evaluate x2 y2 22 dV where E is bounded below by the cone 4 7r6 and above by the sphere E F 9quot th 01 27r 7r6 2 Comment Look at p p2 sin 4 dpd4dl9 0 0 0 Evaluate M952 y2 dV where E is bounded below by the cone 4 7r6 and above by the sphere p 2 E 27r 7r6 2 Comment Look at p sin 4 p2 sin 4 dpd4dl9 0 0 0 Evaluate de where E is bounded below by the cone 4 7r6 and above by the sphere p 2 E 27r 7r6 2 Comment Look at p sin 4 cos 9 p2 sin dpd4d9 0 0 0 Find the mass of a solid hemisphere H of radius 2 if the density at any point is proportional to its distance from the center of the base 27r 7r2 2 Comment Look at p p2 sin 4 dpd4dl9 0 0 0 Find the volume of the solid bounded above by the sphere as y2 22 4 laterally by the cone 2 me 32 and below by the sphere x2 y2 22 1 27r 7r4 2 Comment Look at p2 sin4dpd4d9 0 0 1 Convert the given triple integral into spherical coordinates 3 W m a Z x2 y2 22 dzdydx 73 7W 0 3 M m b x2 y2 22dzdxdy 0 0 x522y2 27r 7r2 3 Comment a p cos 4 p p sin 4 dpd4dl9 0 0 0 b The solid is the portion of the the sphere x2 y2 22 18 that lies both inside the cylinder 952 y2 9 7r2 7r4 2 3 and the cone 2 me 32 that all lies in the rst octant p2 sin4dpd4d9 0 0 3 sin 4

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