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Finding Volume of Revolution with Multivariate Calculus
For some function f(x) it is possible to rotate it along the x-axis and find the area using
$\underset{a}{\overset{b}{\int <!--\; \int \; -->}}\mathrm{\pi <!--\; \pi \; -->}{\left(f\left(x\right)\right)}^{2}dx$
I'm curious how to do this with multivariate calculus. If I represent this in a parametric form, like
$g\left(t\right)=(t,f\left(t\right),0)$
and then rotate using $\left[\begin{array}{ccc}1& 0& 0\\ 0& cos\mathrm{\theta <!--\; \theta \; -->}& -<!--\; -\; -->sin\mathrm{\theta <!--\; \theta \; -->}\\ 0& sin\mathrm{\theta <!--\; \theta \; -->}& cos\mathrm{\theta <!--\; \theta \; -->}\end{array}\right]$
The result is $\mathrm{\Psi <!--\; \Psi \; -->}(t,\mathrm{\theta <!--\; \theta \; -->})=(t,f\left(t\right)\left(cos\mathrm{\theta <!--\; \theta \; -->}\right),f\left(t\right)\left(sin\mathrm{\theta <!--\; \theta \; -->}\right))$ over $a\le <!--\; \le \; -->t\le <!--\; \le \; -->b$ and $0\le <!--\; \le \; -->\mathrm{\theta <!--\; \theta \; -->}\le <!--\; \le \; -->2\mathrm{\pi <!--\; \pi \; -->}$.
I think to compute the volume, I need to use the double-integral $V=\iint <!--\; \iint \; -->⟨\begin{array}{cc}\frac{\mathrm{\partial <!--\; \partial \; -->}\mathrm{\Psi <!--\; \Psi \; -->}}{\mathrm{\partial <!--\; \partial \; -->}t}& \frac{\mathrm{\partial <!--\; \partial \; -->}\mathrm{\Psi <!--\; \Psi \; -->}}{\mathrm{\partial <!--\; \partial \; -->}\mathrm{\theta <!--\; \theta \; -->}}\end{array}⟩\mathrm{\partial <!--\; \partial \; -->}t\mathrm{\partial <!--\; \partial \; -->}\mathrm{\theta <!--\; \theta \; -->}$
where I'm using ⟨⟩ for matrix determinant out of simplicity (and leaving out the bounds).
I can find $\frac{\mathrm{\partial <!--\; \partial \; -->}\mathrm{\Psi <!--\; \Psi \; -->}}{\mathrm{\partial <!--\; \partial \; -->}t}=(1,f^{\prime }\left(t\right)\left(cos\mathrm{\theta <!--\; \theta \; -->}\right),f^{\prime }\left(t\right)\left(sin\mathrm{\theta <!--\; \theta \; -->}\right))$ and $\frac{\mathrm{\partial <!--\; \partial \; -->}\mathrm{\Psi <!--\; \Psi \; -->}}{\mathrm{\partial <!--\; \partial \; -->}\mathrm{\theta <!--\; \theta \; -->}}=(0,f\left(t\right)(-<!--\; -\; -->sin\mathrm{\theta <!--\; \theta \; -->}),f\left(t\right)\left(cos\mathrm{\theta <!--\; \theta \; -->}\right))$.
But the problem is that they would get put in a $3\times <!--\; \times \; -->2$ matrix, so it doesn't make sense to take the determinant.

Finding the volume of a solid bounded by curves.
The region bounded by the given curves is rotated about the specified axis. Find the volume V of the resulting solid by any method.
$x=(y-9{)}^2,x=16;\text{about }y=5$
I used the washer method in terms of y and got
$V=\mathrm{\pi <!--\; \pi \; -->}{\int <!--\; \int \; -->}_5^{13}{16}^2-<!--\; -\; -->(y-<!--\; -\; -->9{)}^{2}dy=\frac{8192\mathrm{\pi <!--\; \pi \; -->}}{5}\text{ which is wrong}$
Also, I am having problems with another similar problem:
The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method.
$x=1-y^4,x=0,\text{ about the line }x=5$
Any help on how to properly set up these integrals would be great, thank you.

Is there any method of finding volume in a high dimensional cube between two planes?
I want to know the volume of this region:
In ${\mathbb{R}}^n$, ($x_1,x_2,...,x_n$)
$0<x_i<1$ for $i=1,2,\dots <!--\; \dots \; -->,n$ (just a cube), and
$0<{\mathrm{ϵ<!--\; ϵ\; -->}}_1<x_1+x_2+\cdots <!--\; \cdots \; -->+x_n<{\mathrm{ϵ<!--\; ϵ\; -->}}_2<n$ (two planes).
Is there any method or formula to get this volume??

Find the volume of the solid in ${\mathbb{R}}^3$.
I need to find the volume of the solid in ${\mathbb{R}}^3$. It is bounded by the following: $y=x^2$, $x=y^2$, $z=x+y+21$ and $z=0$.
I known that the volume is expressed as follows:
$\iiint <!--\; \iiint \; -->1dV$
I have trouble finding the correct limits of integration for this problem.

Finding the volume of the intersection region of two spheres
We have two spheres whose radiuses are a cm each and their centre coordinates are (0, a, 0) and (0, 0, 0) consecutively.
How can we determine the volume of the intersection regions of these given two spheres by using triple integrals?

Expressions of volume and surface area
Now I understand that the volume of a cuboid is $V=whd$ but what I am struggling with is the following:
If w and d are both equal lets call them x, then how can I find an expression for h in terms of x? The closest I can get is
$h=V/x^2$
Is this correct?
In continuation how would I go about finding the surface area A in terms of x only using the answer from the first question?

Finding volume of the solid of revolution
The function is:
$y=\sin <!--\; \; -->(x^2)$ and the boundaries are $y=0$, $x=0$, and $x=\sqrt{\mathrm{\pi <!--\; \pi \; -->}}$.
The function is rotated around the y-axis, and since the functions are difficult to express in terms of y and integrate with respect to dy, I use the formula of $V=2\mathrm{\pi <!--\; \pi \; -->}{\int <!--\; \int \; -->}_0^{\sqrt{\mathrm{\pi <!--\; \pi \; -->}}}\sin <!--\; \; -->(x^2)dx$
My two questions here are:
1) What method can I use to solve the integral of $\sin <!--\; \; -->(x^2)$?
2) How do I limit the boundary in some way to not include the area below the x-axis?

Double Integrals and finding the volume of a pool
I am trying to find the volume of a pool and Im a little rusty on the calculus. How would I go about finding the volume of pool that is Length 10ft by Width 8ft? The depth goes from 3 to 6 feet with a slope of 3ft/10ft.

First word on each volume of a dictionary
Suppose you have to write a dictionary of a language with 27 letters (Spanish): 5 vowels and 22 consonants (yes, Ñ is included). Each word of this new language is made of two letters one wowel and one consonant. Find what would be the first letter on each volume, knowing that the order is alphabetical and there will be 4 volumes.
Knowing how many words there will be in total (220) and in each volume (55) is an easy task, finding the first word of each book seems to be not straightforward (at least to me). I want to know if there is an easy method other than writing the dictionary.

Volume of a solid rotated around axis
The task is finding the volume of the solid found by rotating
$x=2y-<!--\; -\; -->y^2$
From $x=0$ around the axis $y=-<!--\; -\; -->1$

Computing volume of a set in ${\mathbb{R}}^3$
Let $\mathrm{\Omega <!--\; \Omega \; -->}=\{(x,y,z):x^2+y^2\le <!--\; \le \; -->z^2\le <!--\; \le \; -->4\}$.
So, the problem is to find the volume of this set. MY though: comput ${\iiint <!--\; \iiint \; -->}_{\mathrm{\Omega <!--\; \Omega \; -->}}dV$
But, im having trouble seeing the picture of the set, and hence finding limit of integration. Can someone help me?

Finding The Volume Of a Shape That Is Given By the Formula $3x^2+2y^2+z^2\le <!--\; \le \; -->6$
How can I find the volume of the shape that is given by the formula
$3x^2+2y^2+z^2\le <!--\; \le \; -->6$

Finding volume triple integral
Find the volume of the solid enclosed on the outside of the sphere $r=2$ and on the inside by the surface $r=1+\cos <!--\; \; -->\mathrm{\varphi <!--\; \varphi \; -->}$.

Finding volume of pyramid in ${\mathbb{R}}^3$.
Assume we have a pyramid (triangular base) sitting in ${\mathbb{R}}^3$, i.e. we have the coordinates for the vertices A,B,C and D. How can we find the volume of the pyramid without using either its height or its base area and no advanced tools like integration?

Find the volume bounded by the following surfaces
I need to find the volume bounded by the following surfaces:
$x^2+y^2=2z$
$x^2+y^2=3-<!--\; -\; -->z$
I don't know how to proceed in solving this exercise, all I could think of was trying to make a system of these 2 and get $z=1$. I just wanna understand how am I supposed to proceed in finding the boundaries, no need to evaluate the integral.

Intuitive error in finding Volume of sphere using single integration.
To calculate the volume of a sphere of radius R, I considered a thin disc of radius $r=\mathbf{R}sin\mathrm{\theta <!--\; \theta \; -->}$, where $\mathrm{\theta <!--\; \theta \; -->}$ is the angle radius vector on the circumference makes with the axis of the disc. However I considered the volume $dV=$ $\mathrm{\pi <!--\; \pi \; -->}{r}^2\mathbf{R}d\mathrm{\theta <!--\; \theta \; -->}$, $\mathbf{R}d\mathrm{\theta <!--\; \theta \; -->}$ being the infinitesimal thickness of the disc, which is erroneous. The volume of the sphere comes out to be $\frac{3{\mathrm{\pi <!--\; \pi \; -->}}^2{\mathbf{R}}^3}{2}$. The correct thickness is $\mathbf{R}sin\mathrm{\theta <!--\; \theta \; -->}d\mathrm{\theta <!--\; \theta \; -->}$ which is found usually in other derivations gives the correct volume. However I do not understand what exactly is the issue in my approach and why the component of the curve length is considered.

Trouble finding the volume using the shell method
A region is bounded by the line $y=3x+4$ and the parabola $y=x^2$ and is rotated about the line $x=4$.
First I found the limits of integration by finding the points of intersection. They are (-1,1) and (4,16). I then found the radius from the axis of rotation to be $r=4-<!--\; -\; -->x$ and the height to be $h=3x+4-<!--\; -\; -->x^2$.
I used the Volume formula ${\int <!--\; \int \; -->}_a^{b}2\mathrm{\pi <!--\; \pi \; -->}(radius)(height)dx$.
$V={\int <!--\; \int \; -->}_{-<!--\; -\; -->1}^{4}2\mathrm{\pi <!--\; \pi \; -->}(4-<!--\; -\; -->x)(3x+4-<!--\; -\; -->x^2)dx$
$2\mathrm{\pi <!--\; \pi \; -->}{\int <!--\; \int \; -->}_{-<!--\; -\; -->1}^4(x^3-<!--\; -\; -->7x^2+8x+16)dx$
Evaluating this integral I get the volume to be $280\mathrm{\pi <!--\; \pi \; -->}-<!--\; -\; -->{\displaystyle \frac{1055}{6}}$ but this is no where close to the actual volume. I believe that both my radius and height are right and that the formula is right. I've redid the problem a couple of times and ended up with the same answer, what am I missing here?

Finding the volume of a ${\mathbb{R}}^3$ triangle
I have a triangle ABC defined with the points $A=(2.4,-<!--\; -\; -->5.4,6)$, $B=(0,1.1,3.2)$, $C=(-<!--\; -\; -->7.6,3,0)$.
And I'm asked to find the volume of the solid of ${\mathbb{R}}^3$ given by the points between the plane that is made by the points of the triangle and the plaze $z=0$ (The same points of the triangle with all the $z=0$)
I think I should use integrals but I don't know from which regions to do the triple integration.

Multivariable calculus double integration volume question
Use a double integral to find the volume of the solid bounded by graphs of the equations given by:
$\begin{array}{rl}z=x{y}^2,\text{ where: }& z>0\\ & x>0\\ & 5x<y<2\end{array}$
My problem is finding the limits of integration. I know my $f(x,y)$... my guess is my y integration is going from 5x going to 2. But how do I find the x limitations?

Finding the volume of a rotating cross section.
I have to find the volume of a bounded region as it rotates around the Y-axis. My points are $y=6;x=-<!--\; -\; -->2;x=2$; and $y=0$. What would be the steps to solving this and the solution?