Question

1. (a) Using cylindrical coordinates, set up an integral to calculate the volume of the region inside the ellipsoid 4.r? + 4y + z2 = 1 and above the plane z = -1/2.

Answer 1

Rough diagram on the area between ellipsoid bounded above by xy-plane and bounded below a) To set up an integral here its given that it is bouned below by below by z=-1/4 We plug in that value in 4x + 4y +z?= 1 44'tay? += 1 4x²+44² + 1/ = 1 art code bere 4x² + 4y² = 32 hence this is a circe with radius r = √3 and we have to find the integral of the valume region which is between i t and = 51-482-452 cz<-> +443) x?ty=82 the limits of t are 1 K 28/1-46*²742) S2 < 1=482 as a is vonzing

. The integral is 219 V3/4 1 - 482 g. dz drde b) Compute integral 27 Bu Vinar? Ti so z r.drde 21 √3/4 V1.482 Sl z rdr de = 25 v Viegls u er or de let 1-47²_t = -8rdradt substituting this - we get - 15,19 1 (4) 40 que

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