Computation of several integrals with Mathematica

Sorry if my question is very easy. I am learning how to perform symbolic integration with Mathematica. While I can handle easy one-dimensional 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.

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

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

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

4. 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!

• You know about ImplicitRegion[] and ParametricRegion[], I take it? Nov 10 '17 at 8:11
• Thanks! It looks like your help will allow me to solve 2 and 4. But I do not see how to use them with additional constrains like $a>b>c>d>0$ in 1 or $y>0$ in 3. Can you please advice? Nov 10 '17 at 8:28
• Since you've already asked some of these as separate questions, do you still need to keep this question around? Nov 11 '17 at 10:34