(Solved):     Question 1 (5 marks): Evaluate \( \iint_{R} 3(x+y) d x d y \) for the region \( R ...

 

 

Question 1 (5 marks): Evaluate \( \iint_{R} 3(x+y) d x d y \) for the region \( R \) as shown in Figure 1. (Use polar coordinates). Question 2 (9 marks): Let \( G \) be the region bounded above by the sphere \( x^{2}+y^{2}+z^{2}=a^{2} \) and below by the cone \( z=\frac{1}{\sqrt{3}} \sqrt{x^{2}+y^{2}} \). Express \( \iiint_{G}\left(x^{2}+y^{2}\right) d V \) as an iterated integral in a) spherical coordinates b) cylindrical coordinates c) Cartesian coordinates (Do not solve the integrations) Question 3 (5 marks): A thin plate \( (R) \) occupies the smaller region cut from the ellipse \( x^{2}+4 y^{2}=12 \) by the parabola \( x=4 y^{2} \) and the plate's density is \( \delta(x, y)=5 x \). Sketch the region of the plate and find the mass of the thin plate. (Hint: Mass \( =\iint_{R} \delta d A \) ). Question 4 (11 marks): The coordinate axes in Figure 2 run through the centroid of a solid wedge parallel to the labelled edges. If the values of \( a=6, b=6 \) and \( c=4 \), show that the equation of the plane at the top of the wedge is \( 24 y+36 z-48=0 \). Hence, find the moment of inertia with respect to \( x \)-axis, \( I_{x}=\iiint\left(y^{2}+z^{2}\right) d z d y d x \) Figure 2