Solve the questions in the assignment
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1. (2 points) Consider β« π(π₯,π¦)ππ΄ =
π
β« β« (5π¦ + 6)
β4βπ₯2
1
β3
β1
ππ¦ππ₯.
(A) Sketch the region, π of the integration above.
(B) Rewrite the integral in the order of ππππ in polar coordinates.
(solution)
2. (2 points) Set-up an integral and compute the integral to find the total mass of a thin semi-circular
disc with radius of 4, whose mass density is proportional to the distance from the center of the disc.
(solution)
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3. (4 points) Let π be the solid bounded by the following two surfaces:
Surface π1: π§ = β6 β π₯
2 β π¦2
Surface π2: π§ = π₯
2 + π¦2
(A) Sketch the solid π.
(solution)
(B)-(C) Use the specified order of coordinates given below to express (NOT evaluate) the volume
of the solid. [You WILL NOT separate integrals unless you have to.]
(B) Cartesian coordinates in the order of ππ¦ππ₯.
(solution)
(C) Polar coordinates in the order of ππππ.
(solution)
(D) Cartesian coordinates in the order of ππ§ππ¦ππ₯.
(solution)