Questions tagged [solid-of-revolution]
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168
questions
2
votes
1answer
90 views
Question about using Washer method
I want to use the Washer method of integration to find the volume of material in a paraboloid of thickness 0.5mm. For the Washer equation
$$V= π∫b[f(x)^2−g(x)^2]dx$$
$f(x)$ is a parabola, and I know ...
1
vote
2answers
29 views
Solid of revolution for area bounded by parabolas
Show that the volume generated by revolving the region in the first quadrant bounded by the parabolas $y^{2} =x$,
$y^{2}= 8x, x^{2}= y, x^{2}= 8y$ about the x axis is $279 \pi /2$
This problem is to ...
0
votes
0answers
14 views
Setting up integrals for calculating the volume of a solid of revolution
Set up integrals (do not solve them) for finding the volume of the solid formed by revolving the region bounded by $y=(x-1)^2 +\frac{7}{4}$ and $y=-(x+1)^2 +\frac{47}{4}$ about:
$A) y=-1$
$B)x=2$
...
0
votes
1answer
16 views
What side of the axes to compute in a solid of revolution question and how to model a question
I'm a bit confused regarding questions involving solids of revolution.
For example, a question involves a polynomal, say, $x^2+2$ is rotated about the y-axis, find the volume for the solid of ...
0
votes
2answers
40 views
Finding the surface area of a region which is generated by revolving a curve around a line
The following problem is from the book, Calculus and Analytical Geometer by Thomas and Finney. It is early on in the book
so I would expect / hope any integral would be easy to solve.
Problem:
Find ...
0
votes
3answers
36 views
Solid of revolution for given area
Find the volume of the solid obtained by revolving the area enclosed by the curve $27ay^{2} = 4(x-2a)^{3} , x$ axis and parabola $y^{2} = 4ax$ about the $x$ axis.
I am not able to find the area ...
0
votes
1answer
18 views
Volume of the solid generated by revolution of the given curve.
The volume obtained on revolving about $x=a/2$, the area enclosed between the curves $xy^{2} = a^{2}(a-x)$ and $(a-x)y^{2} = a^{2}x$ is ......$?$
I've drawn both curves and both intersect at $x=a/2$, ...
0
votes
1answer
24 views
Confusion related to the volume of a solid of revolution
Well, I've done several excercises about calculating the volume of a solid of revolution some time ago and I don't remember how I got the results of some of them.
The first says "Find the volume of ...
0
votes
4answers
42 views
Why is the slope a part of the area of a surface of revolution?
I have previously learned about the volume of a solid of revolution about the x axis and that equation makes sense to me since it's taking the integral between points a and b of the area of a circle ...
1
vote
0answers
14 views
Finding a value to calculate volume and area using integrals
Given $D=[{(x;y) \in \mathbb R^2 : 1 \le y \le ax^2 +1, 0\le x \le 2/a}], 0 \lt a$, let $W$ be the region obtained by rotating $D$ around $Y$ axis.
A) Find the volume of $W$
B) Find, if possible, ...
1
vote
4answers
39 views
Volume generated by rotating around y-axis, curve $y=x^3$ and the lines $y=0$ and $x=2$
Find the volume of the solid generated by revolving about the y-axis the region bounded by the curve $y=x^3$ and the lines $y=0$ and $x=2$
I first found what $x=2$ would be in terms of y.
$$y= (2)^3
...
1
vote
1answer
21 views
Finding value to calculate volume
Given $D=[{(x;y) \in \mathbb R^2 : 1 \le y \le ax^2 +1, 0\le x \le 2/a}], 0 \lt a$, let $W$ be the region obtained by rotating $D$ around $Y$ axis.
A) Find the volume of $W$
B) Find, if possible, ...
0
votes
0answers
27 views
Implicit formula for surface of revolution
I want to find the unit normal for a surface of revolution of the form
$F(t,s)=(r(t)cos(s),r(t)sin(s),z(t))$
where $\gamma(t)=(r(t),z(t))$ is a curve with unit speed and $r(t)>0$.
I know that if ...
0
votes
2answers
28 views
Confusion About Bounds on a Volume of Revolution
I'm having a little bit of trouble understanding how to determine the bounds and manipulate the equations of a complex solid of rotation problem. Here's a prime example of where I struggle:
Taken ...
4
votes
1answer
40 views
Solid of Revolution About y=2
Find the volume of the solid of revolution bounded by $y=x^4$ and $y=1$ rotated about $y=2$.
Here's my attempt:
$\pi\displaystyle\int_{-1}^1(2-x^4)^2dx$
$\pi\displaystyle\int_{-1}^1(4-4x^4+x^8)dx$
...
0
votes
1answer
18 views
Solid of revolution volume for 3d objects
I need to find the volume of the object:
$z = 3 + \cos x + \cos y$, over $x = 0$, $x = \pi$, $y = 0$, $y = \pi$
The only formula I know if integrating $f^2(x)$ and multiplying by $\pi$.
How does it ...
0
votes
1answer
66 views
Who invented finding volumes through revolution?
If this can't be found, which famous mathematician(s) worked on further developing the idea of finding volumes through revolution?
0
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0answers
36 views
Help with calculus conceptual question
Need help for AP calc AB test coming up but am a bit stumped about how to solve this question:
Suppose a region R has an area A and lies above the x-axis. When R is rotated about the x-axis, it ...
1
vote
1answer
28 views
Volume of Solid through Integration
The solid shown below has a semicircular base of 2cm. Vertical cross-sections of the solid perpendicular to the diameter of the semicircle are right-angled triangles, the heights of which are bounded ...
0
votes
1answer
30 views
How to find the coefficients of this Integral
I am encountering a very tricky problem for me and I am not sure the right approach to solve it. It is simple. It is just telling me to fill in the coefficients of the integral to find the volume of ...
0
votes
1answer
42 views
Volume of a curve using integration [closed]
Consider the region $R$ bounded by the curves $y=ax^2+1, y=0, x=0,\space\text{and}\space x=1, \text{for}\space a\geq-1$.
If $V_1(a)$ is the volume of the solid generated when $R$ is revolved about ...
0
votes
1answer
35 views
Volume of a solid revolution
Find the volume of a solid figure generated by rotating the area of the region bounded by,
$$y_1=x^2-4$$
$$y_2=3x+6$$
and the $x$-axis about the $x$ axis.
I tried solving this using the formula,
$$V= ...
1
vote
1answer
18 views
Finding the surface area of two solids of revolution
I have a couple of questions that are similar in nature:
1) I am trying to find the surface area of this when I rotate it around the x-axis. I have $y = \sqrt{5-x}$ when $3 \leq x \leq 5$
Say $dy/dx ...
0
votes
0answers
17 views
Surface area of revolution query
I've been struggling to grasp this concept. As you see, I was able to achieve the arc length asked for in the first example. Now, however, I seem to be in a pinch regarding the surface area. I ...
3
votes
3answers
69 views
Volume of revolution found by rotating the region bounded by $x=y^2$ and $x=1-y^2$ about the line $x=3$
Find the solid of revolution obtained by rotating the region bounded by the curves $x=y^2$ and $x=1-y^2$ about the line $x=3$.
To solve this problem I tried using the washer method with respect to $y$...
1
vote
0answers
28 views
Medial axis of symmetric solid & volume
I have a problem and would appreciate some pointers. In short, I have a solid which is produced by revolving a 2D polyline around an axis, thus producing a solid of revolution. Now:
1) Is there some ...
0
votes
1answer
42 views
Find the volume of the solid obtained by rotating the region about the y-axis; [closed]
Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis.
$y=x^{-5}, y=0, x=1, x=9;$
about the $y$-axis.
How do I solve this when given ...
0
votes
1answer
63 views
How to calculate volume of a right circular cone's hyperbola segment given position of slice?
PROBLEM: I am working on calculating volumes of geometric solids. All shapes have been pretty basic until now. I am bewildered on how to attack the problem of calculating the volume of a slice of a ...
0
votes
0answers
26 views
Finding the volume of a right cylinder in terms of full surface area and a variable
Let S be the full surface area of a right cylinder. Let H be the height of the cylinder and r be the radius of it's base. Let m = H-r. Find the volume V of the cylinder in terms of S and m.
0
votes
2answers
46 views
Find the volume of the solid obtained by rotating the region bounded by the curves $y=x^2$, $x=5$, and $y=0$ about the $x$-axis
Using disks or washers, find the volume of the solid obtained by rotating the region bounded by the curves $y=x^2$, $x=5$, and $y=0$ about the $x$-axis.
How do I solve this when given rotating region ...
0
votes
1answer
51 views
Solid of revolution using washer method(Gives negative answer)
My teacher even knew that the answer should not be negative but it turned out to be negative. The given was y=x^2, y=4x-x^2, revolving about the y-axis. Here are some of the solution presented, I hope ...
0
votes
1answer
54 views
Optimizing the surface area of an ellipsoid [closed]
How can I optimize the surface area of an ellipsoid to make it as large as possible, while keeping volume fixed? After I rearranged the volume of an ellipsoid to solve for one of the radii, I tried ...
0
votes
1answer
41 views
Finding volume of revolution for $y=x^{2}+2$ about $y=-2$ on the interval $[0,2]$
Finding the volume obtained of a curve after revolving it around $y=-2$, using Mathematica gives
$$V=\pi \int_{0}^{2} \left(4+x^{2} \right)^{2}dx=\dfrac{896 \pi }{15}$$
However, in the solution manual ...
0
votes
1answer
53 views
Why doesn't the second method work?
I have been trying to complete this question:
The region R, bounded by the curve with equation $y=\sin(x)$, $0\leqslant x \leqslant \pi$ and the line with equation $y = \dfrac{1}{\sqrt2}$
The ...
0
votes
2answers
45 views
Volume of solid using shell method
I have a homework question that asks the following:
Use the shell method to find the volume of the solid generated by revolving the region bounded by the line $y=3x+4$ and the parabola $y=x^2$
...
0
votes
1answer
55 views
Volume of revolution of a triangle.
I have a triangle on the $xy$ plane whos base's center is $x_1$ away from the $y$-axis horizontally. I want to rotate it around the y-axis and find the volume of the triangular doughnut shape.
My ...
0
votes
2answers
34 views
I don't understand a part of the solution using the method of shells on this problem
\begin{array} { l } { 4 \mathrm { C } - 3 \text { Find the volume of the region } \sqrt { x } \leq y \leq 1 , x \geq 0 \text { revolved around the } y \text { -axis } } \\ { \text { by both the method ...
1
vote
0answers
24 views
Solid of revolution -finding area
When we calculate volume or area of solid generated by revolving we use absolute sign to cut out negative sign of area..but to be function value of x should give only one value..
But how can we ...
3
votes
2answers
85 views
The volume of solid of revolution rotated about the line $y=x$ [closed]
Find the volume of solid of revolution of region between curves $y=\sqrt x$ and $y=x^2$ in $xy-$plane about the line $y=x$. I know the answer, $\pi/30\sqrt 2$, but how we can obtain it? Should we ...
0
votes
3answers
51 views
Volume of Solid of Revolution (Glass)
We are given $y^2/a^2-x^2/b^2=1$, $y>0$ . If we rotate the hyperbola around the $y$ axis the shape is similar to a glass.
What will the volume of water inside the glass be, in order to fill the ...
0
votes
1answer
40 views
Volume of solid of revolution ($\cos(x)$)
Compute the volume of the solid of revolution that results from revolving $f(x)=\cos(x)$ between $x=-\pi/2$ and $x=\pi/2$ around $y=-1$. I know how to do so around the $x$-axis or the $y$-axis, ...
4
votes
3answers
119 views
Simplest Way to Find Volume of Solid of Revolution Around Given Line
Question
I would like to know the simplest way to find the volume of the solid of revolution created by rotating the parabola $y=x^2$ around the line $y=x$ (the shape shown in blue below). I am ...
2
votes
0answers
134 views
How to find volume and surface area of a spindle torus?
I know that you can use the formulas described in Pappus' centroid theorem, detailed here. But does Pappus' centroid theorem hold true for all forms of a torus: ring, horn, and spindle? I found ...
1
vote
2answers
50 views
Volume of Revolution for $y=1-x^2$ and $y=2x$
There's a question on our review sheet that I don't quite understand. It asks for the integral to find a volume of a solid bounded by the graphs $y=1-x^2$, $x=0$, and $y=2x$. The correct answer is $\...
1
vote
2answers
51 views
Find the area of the surface formed by revolving the given curve about $(i)x$-axis and $(i)y$-axis
Q:Find the area of the surface formed by revolving the given curve about $(i)x-axis$ and $(i)y-axis$
$$x=a\cos\theta ,y=b\sin\theta,0\le\theta\le2\pi$$
About $x-$axis is, $S=2\pi\int_0^{2\pi}b\sin\...
0
votes
2answers
428 views
Find the volume of the solid that results when the region enclosed by the given curves is revolved about the $y$-axis
Q:Find the volume of the solid that results when the region enclosed by the given curves is revolved about the $y$-axis:
$$y=\sqrt{\frac{1-x^2}{x^2}}\:(x>0),x=0,y=0,y=2$$
My first problem is I ...
0
votes
1answer
49 views
Finding the surface area of a solid of revolution
I'm given the function
$x=\frac{1}{15}(y^2+10)^{3/2}$
and I need to find the area of the solid of revolution obtained by rotating the function from $y=2$ to $y=4$ about the $x-axis$.
I've tried ...
0
votes
1answer
34 views
Volume of the solid from rotating four curves
Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis.
y=5+1/(x^2), y=5, x=3, x=6; about the x-axis.
I'm not sure how to solve this ...
0
votes
0answers
55 views
Volume by disk or washer
The volume of the solid obtained by rotating the region enclosed by
$y=x^2, x=y^2$
about the line $x=−5$
can be computed using the method of disks or washers via an integral
I am doing it like
$...
0
votes
1answer
120 views
cross section of torus
Is there any particular name for the plane revolved about an external axis to form a torus? I was thinking of "cross section," but that could be taken as a vertical plane cutting the whole torus in ...
