Expert Help with High School Geometry

Finding the volume of this intersection
How to find the volume of this:
The region common to the interiors of the cylinders $x^2+y^2=1$ and $x^2+z^2=1$ and the first octant.
I tried finding the volume via double integration and the integration gets too complicated. I want to know how to solve this via triple integration. Usually there are x, y, z′s in both equations and that I can solve with triple but I dont know what to do here. I started of doing $x^2+z^2=x^2+y^2$ so $z=y$ but dont know what to do with this it doesnt look like the line of intersection is $z=y$ in the diagram given.

Is there any way to calculate harmonic or geometric mean having probability density function?
I have probability density of function of some data (it's triangular.) How can I calculate harmonic or geometric mean of the data? I know for calculating arithmetic mean of a variable like K, I have to calculate ${\int <!--\; \int \; -->}_0^{\mathrm{\infty <!--\; \infty \; -->}}K.P(K)dK$ but I don't have any ideas for other types of averaging methods (Harmonic and geometric).

Find the volume of the regions enclosed by $z=x^2+y^2-<!--\; -\; -->2$ and $z=30-<!--\; -\; -->x^2-<!--\; -\; -->y^2$
I set up a triple integral with the bounds of the inmost as $x^2+y^2-<!--\; -\; -->2$ to $30-<!--\; -\; -->x^2-<!--\; -\; -->y^2$. The two outer integrals both had the bounds from -4 to 4. When I solved it I got 1024 as the volume, but this isn't correct. Can someone please show me the steps to finding the bounds, and the correct answer?

Distance between (-6,3,4) and (-5,-1,1)?

What is the distance between (-2,-9,10) and (22,5,-6)?

Volume integral - finding the region
I need to find the volume of an object set with the following function:
$x+y+z=1$
And all three axis.
So I converted the function into $z=-<!--\; -\; -->x-<!--\; -\; -->y+1$, and it gave me kind of a clepsydra, crossing x,y plane in $x=y=1$.
So the region seems to be the circle $x^2+y^2=1$.
However, changing x to r, and y to $\mathrm{\varphi <!--\; \varphi \; -->}$ gives me wrong result $\mathrm{\pi <!--\; \pi \; -->}$, and it should be 1/6.
Is the region I've came up with correct?

What is the distance between (1,6) and (7,4)?

How do you find the perimeter of a triangle whose coordinates are X(0,2), Y(4,-1), Z(-2,-1)?

Distance between (8,2) and (-5,13)?

Finding value to calculate volume
Given $D=[(x;y)\in <!--\; \in \; -->{\mathbb{R}}^2:1\le <!--\; \le \; -->y\le <!--\; \le \; -->a{x}^2+1,0\le <!--\; \le \; -->x\le <!--\; \le \; -->2/a],0<<!--\; <\; -->a$ let W be the region obtained by rotating D around Y axis.
A) Find the volume of W
B) Find, if possible, the a values $\mathrm{ϵ<!--\; ϵ\; -->}(0;+\mathrm{\infty <!--\; \infty \; -->})$ so that the volume of W is a minimum, and a maximum
C) Find, if possible, the a values $\mathrm{ϵ<!--\; ϵ\; -->}(1/3;3)$ so that the volume of W is a minimum, and a maximum
Well, what I've done so far is finding the inner and outer radius to calculate the volume in terms of Y. The inner radius would be be $\sqrt{\frac{y-<!--\; -\; -->1}{a}}$ and the outer would be $\frac{4+a}{a}$, which is a evaluated in the parabola. Then, the volume would be ${\int <!--\; \int \; -->}_1^{\frac{4+a}{a}}(\sqrt{\frac{y-<!--\; -\; -->1}{a}}{)}^2-<!--\; -\; -->(\frac{4+a}{a}{)}^{2}dx$.
And that's the function that I have to differentiate to find its maxima and minima, which, after differentiating, is 1/a. Is ok what am I doing? How can I go on?

Distance between (7,-46,1) and (7,-24,1)?

How can the sin/cos/tan values in a right angle be negative?
A and B are both obtuse angles such that $\sin <!--\; \; -->(A)=\frac{5}{13}$ and $\tan <!--\; \; -->B=\frac{-<!--\; -\; -->3}{4}$. Find exact values for $\sin <!--\; \; -->(A+B)$.
Assuming that the pythagorean theorem is used to answer the question, how is it possible that the values for $\tan <!--\; \; -->B=\frac{-<!--\; -\; -->3}{4}$? (That is, a 3-4-5 right angle triangle... how can a side be negative?)

Finding the height and top radius of cone so that volume is maximum and Finding the angle so that the volume is maximum
You have a 6-inch diameter circle of paper which you want to form into a drinking cup by removing a pie-shaped wedge with central angle theta and forming the remaining paper into a cone. - You are given that 3 is the slant height
a) Find the height and top radius of the cone so that the volume of the cup is as large as possible.
b) What is the angle theta of the arc of the paper that is cut out to create the cone of maximum volume?
I know how to do related rates with volume, but I can't seem to figure it out with angles being cut out from a circle.

Distance between (-4,0,2) and (2,-3,1)?

Suppose $R$ is a partial order on a set $A$, and $B\subseteq <!--\; \subseteq \; -->A$.
Prove that, if $R$ is a total order and $b$ is a minimal element of $B$, then $b$ is the smallest element of $B$.

Geometric or Poisson
Drought length is referred to as the number of consecutive time intervals in which the water supply remains below a critical value. Consider the drought length as a random variable, denoted as Y, which is assumed to have a geometric distribution with $p=0.409$
1. What is the probability that a drought lasts exactly 3 intervals? (0.0844)
2. What is the probability that a drought lasts at most 3 intervals? (0.878)

Prove that the intersection angle between the Simson lines of two triangles inscribed in the same circle it's the same for any point.
Suppose the triangles ABC and DEF share the circuncircle C, and P and Q are any diferent points on C. Let l, m be, the Simson lines of P related to ABC and DEF, and p, q the Simson lines of Q related to ABC and DEF, then i must prove the angle between l and m equals the angle between p and q.
I just have one Theorem about the Simson line:
Theorem: Let P, Q be two points on the circuncircle, C, of the triangle ABC. Let l, m be their respective Simson's lines. Then the angle between l an m equals to the half of the central angle POQ, where O is the center of C.

Distance between (8,-7,-4) and (9,2,2)?

How to find an angle between the extension of a diameter and a chord in a semicircle

O is the center of the semicircle, if $\mathrm{\angle <!--\; \angle \; -->}EOD={45}^{\circ <!--\; \circ \; -->}$ and $OD=AB$, find $\mathrm{\angle <!--\; \angle \; -->}BAC$

Distance between (-2,0,1) and (0,4,-2)?