Practice Problems with Solution - Calculus IV | MTH 254, Study notes of Advanced Calculus

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Math 254 - Practice Problems Wed, Nov 16 Justify your reasoning. Change of Variables 1. Let R be the region bounded the lines e+ y = 1la—y=—-la-y=3,2—-y=1. Using an appropriate change of variables, evaluate If A(a — y)sin(a — y) dA. Don’t R forget the Jacobian! 2. Using 7 = psingcosd.y = psindsin?, z = pcos®@, prove that the Jacobian for switch- ing a triple integral to Spherical coordinates is seat = p’sing. 3. Using ¢ = rcosé,y = rsin@,z = 2, aprove that the Jacobian for switching a triple integral to Cylindrical coordinates i is 2 1 v kg v | Coy] | ° y_¥ —~ | 2 (uw) % yl” oa fy 7 io ’ t J lersdsrnley A 5 { dusmv[- ‘aldvdy i “ Xe Vo seomeesnigenemetoat eh (uv a) |eu® D o(xyt) ,2) |: sme) - esmgsmd Per@end z-¥) 5g “_ : sindsmO grinder gem? sm =Cos@ Cg sagen sat is cu sy cos 0 -esn@ 2 g’snecne ave + PImP(Q smeorO a> Sing (oz6sivt) * a“ — e" sme Cos*ep (sim +c078)— 0 ~ orsrnee (cos ag + sre) - cos -vsind O sao rend © = royd enti) 0 O |