Sorry if my question is very easy. I am learning how to perform symbolic integration with Mathematica. While I can handle easy onedimensional integrals, I am not able to handle more complex problems. My problem is that I do not understand how to describe integration domains and how to use additional information about them. Please show me how to do it.

Compute the volume of subset of 4 dimensional ellipsoid ($ a>b>c>d>0$ ) $ $ \frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}+\frac{t^2}{d^2} \leq 1,\ \frac{x}{a}+\frac{y}{b}+\frac{z}{c}+\frac{t}{d} > 1 .$ $

Compute surface integral $ \iint x\ dS$ . The surface is given parametrically $ x=3t+1, y=a^3 \sin t, z=a^3 \cos t, a \in [\frac12, 1], t \in [\frac\pi6,\frac\pi4]$ .

Compute second order integral over “top” surface of a hemisphere $ x^2+y^2+(z+1)^2 = R^2, y>0$ $ $ \iint dz \ dx + y^2z \ dz\ dy.$ $

Surface $ S$ is given by equation $ z=x \sin y, x \in [0, 1], y\in[0, \pi]$ . Compute integral $ \int xy \ dy$ over the border of $ S$ . $ S$ is oriented counterclockwise if you look at it “from top”.
Even partial answers are highly appreciated. Thanks a lot for your time!