6. (i) (a) Rewrite the integral SSR f(x, y)dxdy in different coordinates u, v. (b) Hence...

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6. (i) (a) Rewrite the integral SSR f(x, y)dxdy in different coordinates u, v. (b) Hence derive the form the integral R f(x, y)dxdy will take in polar co- ordinates. (ii) (a) A new modern sculpture is to be displayed at Durham Science Site. It is given by the region bounded by the two surfaces in (x, y, z) space defined by z = (x + y - 1) and z = 4. What is the volume of the sculpture? (b) What is its centre of mass assuming it has constant density? C 6. (i) (a) Rewrite the integral SSR f(x, y)dxdy in different coordinates u, v. (b) Hence derive the form the integral R f(x, y)dxdy will take in polar co- ordinates. (ii) (a) A new modern sculpture is to be displayed at Durham Science Site. It is given by the region bounded by the two surfaces in (x, y, z) space defined by z = (x + y - 1) and z = 4. What is the volume of the sculpture? (b) What is its centre of mass assuming it has constant density? C