Module MA1132 (Frolov), Advanced Calculus Tutorial Sheet 7 To be solved during the tutorial session Thursday/Friday, 10/11 March 2016 Name: You may use Mathematica to sketch the integration regions and solids, and to check the results of integration. 1. Evaluate the double integral ZZ p 4x2 − y 2 dxdy , I= R where R is the region enclosed by the lines y = 0, y = x, and x = 1. 2. Sketch the integration region R and reverse the order of integration (a) 4 Z 12x Z f (x, y)dydx (1) f (x, y)dydx (2) 3x2 0 (b) Z 0 1 Z 3x 2x 3. Sketch the integration region R and write the expression as one repeated integral by reversing the order of integration Z 1Z y Z 3Z 1 f (x, y)dxdy + f (x, y)dxdy (3) 0 y 2 /9 1 y 2 /9 4. Find the volume V of the solid bounded by (a) the planes x = 0, y = 0, z = 0, the cylinder x2 + y 2 = a2 and the hyperbolic paraboloid z = xy in the first octant. (b) the planes y = 0, z = 0, y = ab x, and the elliptic cylinder 1 x2 a2 + z2 c2 = 1.