Reference no: EM132388318

Questions -

(1) Find the volume of the solid under the surface z = xy and above the triangle in the x-y plane with vertices (1, 1), (4, 1) and (1, 2).

(2) Evaluate the integral ∫∫_R(x+y)dA, where R is the region between the circles x² + y² = 1 and x² + y² = 4 defined only for x ≤ 0. In this case the answer should be negative. Interpret this result.

(3) Consider a two dimensional lamina with constant density occupying the region in the x-y plane shown in the following diagram

Find the centre of mass of the lamina, and determine for which range of L values the centre of mass lies inside the region, and for which range of L values the centre of mass lies outside the region. (In your answer, assume L < 0, as indicated in the diagram).

(4) Express the integral

₀∫¹₀∫^z_y^2∫¹f(x, y, z) dx dy dz

in the order dz dx dy.

(5) Consider a solid cylinder of radius R and a spherical shell with outer radius R and inner radius a. Both objects have the same constant density. Find the height of the cylinder such that the two objects have the same moment of inertia when rolling down a slope.

(6) Let C be a circle of radius 1 centred at the point (x, y) = (0, 1), T the unit tangent vector to C traversed in an anticlockwise direction, and n the unit normal vector to C directed away from the centre of the circle. Consider the vector field F(x, y) = xi + xyj.

(a) Calculate F · T ds.

(b) Calculate F · n ds.