Math 324 Practice Midterm: Surface Area, Mass, Moment of Inertia, Integrals, and Variables, Exams of Calculus

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1. Find the surface area of the surface D = {(x, y)|x (^2) + y (^2) ≤ 1 , −x ≤ y ≤ x z} (^) .= 12 (x^2 + y^2 ) over the region

2. Suppose we wish to make a thin metal washer with constant density,outer radius 2 cm., and inner radius a cm. for some constant a, 0 ρ< a < g/cm.^2 2., with We represent the washer by the region in the x, y plane, D = {(x, y)| a^2 ≤ x^2 + y^2 ≤ 4 }. (a) Compute a formula for the mass of the washer, m, in terms of a and ρ. (b) Compute a formula for the moment of intertia of the washer about thein terms of a and ρ. y-axis, Iy, (c) If we want m = 9 g. and Iy = 13 g.cm.^2 , what value must we choose for a?

4. Let S denote the surface defined by the equation x^2 + y^2 + z = 4 above the xy-plane. (a) Find a parametrization for S. (b) Evaluate: (^) ∫ ∫ S^ √1 + 4^1 x^2 + 4y^2 dS.

5. The hyperbolic sine and cosine functions are given by: cosh t = et^ + 2 e−t, sinh t = et^ − 2 e−t (Notice cosh x = r cosh t,^2 ty − = sinh r sinh^2 t = 1) The object of this problem is to use a change of variables, t, to compute ∫∫ D x − y dA, where D = {(x, y)| 0 ≤ y, y < x, x^2 − y^2 ≤ 1 }. (a) Compute the Jacobian ∂ ∂((x,yr,t)) (b) Describe the region that corresponds to(Solution: D = {(r, t)| 0 < r ≤ 1 , 0 ≤ t < D in the∞}) r, t plane. (c) Use change of variables to compute ∫∫ D x − y dA.