[Math] How to rotate function around x axis

calculusintegration

The problem is:

Revolve the function around the $x-$axis, then find the volume enclosed by the $3D$ shape from $x_1=0$ to $x_2=16$.

The following formula may be used to determine the volume of the solid:

$$\displaystyle V=\int_{x_1}^{x_2} \pi[f(x)]^2dx$$

The function is as follows:

$$f(x)= \dfrac{6.4}{x+12} \sin\left(\dfrac{2~\pi}{6.5}x\right)+3$$

My question is:

a) What would the resulting function be?

And more importantly,

b) How would I convert this into a $3D$ function, that is, incorporate $z$ so it can be graphed in $3D$?

Thanks

Best Answer

Your equation for the volume is correct. However, actually plotting the volume is somewhat more complicated. Specifically, in 3D space you need a matrix of terms for each of $x, y$, and $z$. This is most easily visualized in a vertical arrangement, with rotation about the $z$-axis. So imagine the line $r=f(z)$. Then take a vector $\theta\in[0,2\pi]$ with however many points you wish and for each value of $z$ find $X=r\cos\theta$, $Y=r\sin\theta$. These are matrices of the size of $r$ by the size of $\theta$. The $Z$ matrix is just a uniformly spaced matrix for all the $z$ and $\theta$ values. Many computer languages have such functions built in. I used Matlab's cylinder function to create the figure below.

EDIT: At the request of the OP, I am adding the Matlab code. Note that function cylinder is a Matlab built-in function described here.

x=16*linspace(0,1,4001)';
f=6.4./(x+12).*sin(2*pi*x/6.5)+3;

[X,Y,Z]=cylinder(f,50);

figure;surf(X,Y,16*Z)
axis equal
shading flat
xlabel('X');ylabel('Z');zlabel('Y','Rotation',0)

Related Solutions

[Math] Revolve a 3D shape around an axis to create a 4D shape (and so on and so fourth)

Building a higher dimensional shape from rotation is an easy process. Starting with a circle , $S^1$ , there are two distinct ways to rotate into 3D. These rotations happen around an n-1 plane, into n+1. Axes that lie on the n-1 plane remain stationary, while leaving one axis over to be transformed mathematically. One can establish the plane of rotation to pick out the single axis "$x_m$", for any shape.

Bisecting Rotate into $x_n$, transforming $x_m$:

$x_m^2 \rightarrow x_m^2 + x_n^2$

Examples:

• circle, $S^1$ :

$x^2 + y^2 = r^2$

Bisecting rotate around axis x into z will transform y, making the 2-sphere:

$y^2 \rightarrow y^2 + z^2$,

$x^2 + y^2 + z^2 = r^2$

Continuing to bisecting rotate will only add dimensions to the $S^n$, leading to higher n-spheres.

• sphere, $S^2$ :

$x^2 + y^2 + z^2 = r^2$,

Bisecting rotate around plane xy into w, will transform z, making the 3-sphere:

$z^2 \rightarrow z^2 + w^2$,

$x^2 + y^2 + z^2 + w^2 = r^2$

Bisecting rotate around plane xyz into v, will transform w, making $S^4$:

$w^2 \rightarrow w^2 + v^2$,

$x^2 + y^2 + z^2 + w^2 + v^2 = r^2$

Bisecting rotate around plane xyzw into u, will transform v, making $S^5$:

$v^2 \rightarrow v^2 + u^2$,

$x^2 + y^2 + z^2 + w^2 + v^2 + u^2 = r^2$

As we can see from this, bisecting rotation of $S^n$ around any hyperplane, m-number of times, will yield $S^{n+m}$ .

Non-Intersecting Rotate into $x_n$, transforming $x_m$:

$x_m \rightarrow (\sqrt{x_m^2 + x_n^2}-a)$

• Performed by shifting the starting shape by a certain value of -a , then sweep around the n-1 plane, into n+1, to make a toroidal ring.

circle : $x^2 + y^2 = r^2$

shift the circle by -a along y, then sweep around x, into z:

$y \rightarrow (\sqrt{y^2 + z^2} -a)$,

$x^2 + (\sqrt{y^2 + z^2} -a)^2 = r^2$

and re-orienting the shape,

$(\sqrt{x^2 + y^2} -a)^2 + z^2 = r^2$,

we get the equation for a torus, $T^2$. For $S^1$, we're not confined to only transforming y , as x can be used equally so.

Building 4D Shapes

• Starting with either $S^2$ or $T^2$, we can bisecting or non-intersecting rotate to build 4D shapes. There are two ways to rotate $S^2$ , and four ways to rotate $T^2$. The bisecting rotate for $S^2$ into $S^3$ was already detailed above, so I'll show the non-intersecting rotation:

• $S^2$ :

$x^2 + y^2 + z^2 = r^2$,

shift the sphere by -a along x, sweep along circular path around plane yz, into w:

$x \rightarrow (\sqrt{x^2 + w^2} -a)$,

$(\sqrt{x^2 + w^2} -a)^2 + y^2 + z^2 = r^2$

And, rearranging to

$(\sqrt{x^2 + y^2} -a)^2 + z^2 + w^2 = r^2$,

we get a compact 3-manifold of genus-1, defined as $S^2$ x $S^1$ , sphere bundle over the circle.

The four ways to rotate a torus into 4D:

• Torus, $T^2$:

$(\sqrt{x^2 + y^2} -a)^2 + z^2 = r^2$

Degree-4 Surfaces

1. Bisecting rotate around plane xz (or yz), into w, will transform y:

$y^2 \rightarrow y^2 + w^2$,

$(\sqrt{x^2 + y^2 + w^2} -a)^2 + z^2 = r^2$

reorient to

$(\sqrt{x^2 + y^2 + z^2} -a)^2 + w^2 = r^2$

A 3-manifold of genus-1, $S^1$ x $S^2$ , circle bundle over the sphere.

2. Bisecting rotate around plane xy, into w, will transform z:

$z^2 \rightarrow z^2 + w^2$,

$(\sqrt{x^2 + y^2} -a)^2 + z^2 + w^2 = r^2$

A 3-manifold of genus-1, $S^2$ x $S^1$ , sphere bundle over the circle.

Degree-8 Surfaces

3. Non-intersecting rotate around plane xz (or yz), into w, transforms y. Shifted by -a along y-axis:

• Torus, $T^2$:

$(\sqrt{x^2 + y^2} -b)^2 + z^2 = r^2$,

$y \rightarrow (\sqrt{y^2 + w^2} -a)$,

$(\sqrt{x^2 + (\sqrt{y^2 + w^2} -a)^2} -b)^2 + z^2 = r^2$

reorient to

$(\sqrt{(\sqrt{x^2 + y^2} -a)^2 + z^2} -b)^2 + w^2 = r^2$

A 3-manifold of genus-2, the 3-torus $T^3$, torus bundle over the circle.

4. Non-intersecting rotate around plane xy, into w, transforms z. Shifted by -a along z-axis:

$z \rightarrow (\sqrt{z^2 + w^2} -a)$,

$(\sqrt{x^2 + y^2} -b)^2 + (\sqrt{z^2 + w^2} -a)^2 = r^2$,

A 3-manifold of genus-2, the tiger, circle bundle over the Clifford flat torus. For simplicity and clarity, I'm adopting the notation C2 to represent the Clifford torus, in fiber bundle terms. Not to be mistaken for the space $C^2$. (I hope that's okay with everyone). So, in the case of this 4D hypertorus, we can define it as $S^1$ x C2 , which is homeomorphic to $T^3$, as the product of three circles, but uniquely different.

So, to recap 4D:

$S^3$ : $x^2 + y^2 + z^2 + w^2 = r^2$

$S^1$ x $S^2$ : $(\sqrt{x^2 + y^2 + z^2} -a)^2 + w^2 = r^2$

$S^2$ x $S^1$ : $(\sqrt{x^2 + y^2} -a)^2 + z^2 + w^2 = r^2$

$T^3$ : $(\sqrt{(\sqrt{x^2 + y^2} -a)^2 + z^2} -b)^2 + w^2 = r^2$

$S^1$ x C2 : $(\sqrt{x^2 + y^2} -b)^2 + (\sqrt{z^2 + w^2} -a)^2 = r^2$

Building 5D Shapes

Now, we can continue this process of rotating 4D shapes into 5D, producing the $S^4$ , and 11 distinct types of hypertoric rings.

The compact 4-manifolds that curve into 5D:

Degree-2 Surface of genus-0

$S^4$ : $x^2 + y^2 + z^2 + w^2 + v^2 = r^2$

Degree-4 Surfaces of genus-1

$S^3$ x $S^1$ : $(\sqrt{x^2 + y^2} - a)^2 + z^2 + w^2 + v^2 = b^2$

$S^2$ x $S^2$ : $(\sqrt{x^2 + y^2 + z^2} - a)^2 + w^2 + v^2 = b^2$

$S^1$ x $S^3$ : $(\sqrt{x^2 + y^2 + z^2 + w^2} - a)^2 + v^2 = b^2$

Degree-8 Surfaces of genus-2

$S^2$ x $T^2$ : $(\sqrt{(\sqrt{x^2 + y^2} - a)^2 + z^2} - b)^2 + w^2 + v^2 = c^2$

$S^1$ x $S^2$ x $S^1$ : $(\sqrt{(\sqrt{x^2 + y^2} - a)^2 + z^2 + w^2} - b)^2 + v^2 = c^2$

$T^2$ x $S^2$ : $(\sqrt{(\sqrt{x^2 + y^2 + z^2} - a)^2 + w^2} - b)^2 + v^2 = c^2$

$S^2$ x C2 : $(\sqrt{x^2 + y^2} - a)^2 + (\sqrt{z^2 + w^2} - b)^2 + v^2 = c^2$

$S^1$ x [$S^2*S^1$] : $(\sqrt{x^2 + y^2 + z^2} - a)^2 + (\sqrt{w^2 + v^2} - b)^2 = c^2$

Degree-16 Surfaces of genus-3

$S^1$ x C2 x $S^1$ : $(\sqrt{(\sqrt{x^2 + y^2} - a)^2 + z^2} - b)^2 + (\sqrt{w^2 + v^2} - c)^2 = d^2$

$T^2$ x C2 : $(\sqrt{(\sqrt{x^2 + y^2} - a)^2 + (\sqrt{z^2 + w^2} - b)^2} - c)^2 + v^2 = d^2$

$T^4$ : $(\sqrt{(\sqrt{(\sqrt{x^2 + y^2} - a)^2 + z^2} - b)^2 + w^2} - c)^2 + v^2 = d^2$

And, getting into 6D, we'll get $S^5$ , and 32 distinct toroidal shapes. The interesting ones are a degree-32 surface, as compact 5-manifolds of genus-4. Moving further, 7D will have $S^6$ and 89 types of hypertorus, some of which are degree-64 of genus-5. What's neat about the combinatoric nesting of diameters, is that it follows rooted trees with nested leaves, the integer sequence A000669. All combinations of bisecting and non-intersecting rotations of a circle exhaust the total possible combinations in each dimension. Part of my self-study and personal research has been this combination sequence, that defines the implicit surface equations of hypertoric shapes. I can elaborate on this further, if anyone is interested.

I have done quite a bit of illustration of these shapes, too:

4D Hypertoric Shapes

5D Hypertoric Shapes

and many of the 6D and 7D, and 9D as well.

[Math] Volume of the solid of revolution generated when the parabola spins around the $x$ axis

Draw a picture. We have a parabola whose axis of symmetry is the line $y=2$, and which opens leftward. The apex of the parabola is at $(4,2)$.

The line $y=1$ divides the part of the parabola to the right of the $y$-axis into two parts, the fat part above the line $y=1$, and a much thinner part below the line $y=1$. It is not clear from the wording which part is being spun. We will assume it is the fat part. Minor modification will take care of things if it is the thinner part.

It is most convenient to use the Method of Cylindrical Shells. Look at a thin horizontal strip, going from height $y$ to height "$y+dy$."

Spin this strip about the $x$-axis. We get a cylindrical shell of thickness "$dy$." The shell has radius $y$ and height $x=4y-y^2$. So the shell has volume approximately equal to $2\pi y(4y-y^2)\,dy$. "Add up" (integrate) from $y=1$ to $y=4$. We get volume $$\int_{y=1}^4 2\pi y(4y-y^2)\,dy.$$

Remark: If we want to integrate with respect to $x$, things get a little more complicated. You did it along those lines, so we write down suitable expressions. The volume from $x=0$ to $x=3$ is equal to $$\int_{x=0}^3 \pi[(2+\sqrt{4-x})^2-1^2]\,dx.$$ To this we must add the volume from $x=3$ to $x=4$, which is $$\int_{x=3}^4 \pi[(2+\sqrt{4-x})^2-(2-\sqrt{4-x})^2]\,dx.$$

Best Answer

Related Solutions

[Math] Revolve a 3D shape around an axis to create a 4D shape (and so on and so fourth)

[Math] Volume of the solid of revolution generated when the parabola spins around the $x$ axis

Related Question