Revolution Definition and 393 Threads

Homework Statement y= -(x/6) + b, find the volume as this solid is rotated 360 degrees around the Y axis Homework Equations If I were given the interval at which I needed to find the volume and/or the value of B I could easily do this using the formula: [pi] Integrate: (R(y))2 dx...

As part of an assignment on Approximating Areas and Volume I am asked to derive the equation shown in the image attached. The question reads: "It can be shown that if y = f(x) is revolved around the x-axis to form a solid between x=a and x=b then the volume of the solid is give by the...

Hey. Thanks in advanced for the help. This site has helped me a lot through the years. Homework Statement Find the volume of the solid formed by rotating the area within y=e^x and y=sin x when 0<x<pi Homework Equations The Attempt at a Solution I've tried it like 10 times on...

Homework Statement A tunning fork with frequency f = 512 Hz is held by someone who swings it vigurously in a circle in the horizontal plane. The radius of the circle is 1.0 m, and the frequency of revolution is 3.0 rev/s. (a) What are the max and the min frequencies that a second person...

Homework Statement Some bacteria are propelled by motors that spin hair-like flagella. A typical bacterial motor turning at a constant angular velocity has a radius of 1.8x10-8m, and a tangential speed at the rim of 1.8x10-5 m/s. (a) What is the angular speed (the magnitude of the angular...

Homework Statement Here is the problem : Find the volume of the solid generated when the area bounded between the following functions rotates along the y=4 line. Functions: y=x^3 y=4x Homework Equations Functions...

Homework Statement The line y = x + 6 meets the curve y = x2-4x at the points P and Q. Find the volume of the solid generated when the area enclosed by the line and the curve is rotated through 360o about the x-axis Homework Equations Integration The Attempt at a Solution I've...

[b]1. A particle of mass m and angular moment L moves in a central force V=(1/2)kr^2 (k>0). Find the period of revolution for the circular movement and the period of small radial oscillations around the stable cicular orbit. Homework Equations The Attempt at a Solution Well I tried...

So at the end of the lengthy thread https://www.physicsforums.com/showthread.php?t=404650", a similar but new question of mine arose: If you consider the Earth rotating, you can measure differences in weight at different points on the Earth (namely, the extremes being the poles and the...

Homework Statement Find the volume of the solid bounded by the curves y = x^{1/3} and y = x when rotated around y=1. Homework Equations Volume with washers: V = \pi \int R(x)^{2}-r(x)^{2} dx where R(x) and r(x) are functions of x defining the inner and outer radii of the washers...

Homework Statement Use the method of cylindrical shells to nd the volume of the solid obtained by rotating the region bounded by the curves y = x4, y = 1, x = 0 about the y-axis. Sketch the region and a typical shell The Attempt at a Solution I am just going to set it up, but I am not going...

Homework Statement Find the area of the region bounded by y=0, x=9, y=x/3 and rotated about y=-2 The Attempt at a Solution The answer is \pi\int^{9}_{0}(2+x/3)-2^2dx I'm just wondering if this is the same thing as saying find the area of the region bounded by y=2 instead of y=0...

Homework Statement Find the area of the surface generated when you rotate the parabola y=x2 0 less than or equal to x less than or equal to the square root of k, around the y-axis. You should end up with a simple formula in terms of the constant k. Homework Equations S=2\pi\intyds...

Homework Statement Find the surface area generated by rotating y=5-4x^(3/2), 0\leq x\leq 1 about x=2.Homework Equations SA = 2\pi\int_{a}^{b}(r\cdot ds)dx The Attempt at a Solution I simply filled in the formula for the given question, and I'm getting stuck at integration time. SA =...

Homework Statement The region bounded by y=e^{-x^{2}}, y=0, x=0, and x=1 is revolved about the y-axis. Find the volume of the resulting solid. Homework Equations integral from a to b of pi*f(y)^2The Attempt at a Solution If y=e^-x^2 and I am revolving about the y-axis then I need to rewrite...

Homework Statement The solid formed when the region bounded by y = x^2 and y = 2 - x^2 is revolved about the x-axis Homework Equations disc method with respect to x-axis the integral of : (pi * (f(x)^2 - g(x)^2)) The Attempt at a Solution When I square each function and...

Hello there. Suppose I have a function: y=3x^{2/3}-1 I want to find the surface area of the solid formed when the part of the curve between x=0 and x=8 is revolved about the x-axis.The curve crosses the x-axis at a point (1/3)^{3/2} The derivative of the function is...

Hello, I was wondering if anyone could help me with deriving the volume created by the rotation of a polar equation around the initial line. So, I thought about adding the surface area of cones (multiplied by d\theta) if each cone the triangle created with s-length of f(\theta) and r-length...

Single Var Calculus -- Volumes of Revolution consider the curves y = 6 x = 0 and y = x2+2 Revolve the bound area around the y-axis and find the volume of the product solid. Here's what I did. r = x = (y - 2)1/2 V = pi * INTEGRAL from y = 2 -> y = 6 of r2 dy = y - 2 dy = y2/2 -...

Homework Statement The combination of an applied force and a friction force produces a constant total torque of 35.0 N · m on a wheel rotating about a fixed axis. The applied force acts for 5.90 s. During this time, the angular speed of the wheel increases from 0 to 10.1 rad/s. The applied...

Homework Statement The height of an (axially symmetric) pipe z as a function of the distance from the axis of symmetry is z = 2-2x^{2} , where both z and x are measured in metres, and where 0 \leq z \leq 2 and 0 \leq  x  \leq 1 What is the total volume of hay in cubic metres in...

Homework Statement Find the volume of the solid generated by revolving the region bounded by the given graph about the indicated line using any method. y=6-2x-x2 y=x+6 About the axis x=1 Homework Equations Shell method: 2*pi * INT[ p(x) * h(x) ] dx The Attempt at a...

I tried doing just the integration and I was totally stumped, and Wolfram-Alpha has been no help at all Find the surface area obtained by rotating about the x-axis: ln(sec x) for 0 < x < (pi/6) This problem appeared on an exam I had, but all that was required was to set up the...

Hi, A cylinder is to be coated on a rotating lathe, with a coating rod moving by a mechanical robot in the X direction (back and forth). The cylinder is 300mm in length, 19mm in diameter. What is the distance traversed per revolution of a rotating cylinder, with the lathe rotating at 1200RPM...

Dark matter is believed to have a more homogenous distribution than normal matter, a bias of at least 2 is used in models of large-scale structure in order to justify large voids and superclusters in a homogenous universe. If Dark matter is so evenly distributed I have some difficulties to...

Homework Statement Consider a vertical cone of height h whose horizontal cross-section is an ellipse and whose base is the ellipse with major and minor semi-axes α and β. Verify that the volume of the cone is παβh/3. [ Hint: The area of an ellipse with major and minor semi-axes α and β is...

Homework Statement 2\pi \int_0^6 y\sqrt{25- (y- 1)^2}dy + 5\pi That's the integral i need solved 2. The attempt at a solution[/b] so first i subbed u=y-1 took the 2 pi out of the integral that got me 2 integrals u*sqrt(25-(u)^2) du + sqrt(25-(u)^2) du the first integral =...

1. Homework Statement k so here is the equation i need help with that will find me the volume of a sphere 2*pi*y*sqrt(25-(y-1)^2) dy - 5*pi from 0 to 6 the 5 pi is the volume of a cylinder 2. Steps so first i subbed u=y-1 took the 2 pi out of the integral that got me 2 integrals...

I've simplified my question to make it easier to explain; I'm trying to find a way to determine the motion of a freely turning body as resultant of 2 counter-weights on rotating arms. Here's my write-up: Description: Electric motors turn both orange shafts clockwise at constant angular...

Can anyone confirm if I have done the following work correctly Find the volume of a solid of revolution obtained by rotating about the y-axis the region bounded by y = the fifth root of x and 2x^2 - 3x + 2. By drawing the graph, I figured out that I need to use the method of cylindrcal...

Homework Statement Hi, well, I don't have to solve any problem yet. I have an inquietude about solids of revolution. I've been reading this: http://en.wikipedia.org/wiki/Solid_of_revolution and this: http://en.wikipedia.org/wiki/Disk_integration What I wanted to know is if its the same this...

Find the volume of the solid st, 1. y=cos x , y= 0 in [0,pi] ; Rotated around x=1 2. I am slightly confused, I see that the area will double around twice so I can just use the left half of the curve. I am just not sure how to do so.

Homework Statement let R be the region enclosed by the x-axis,y-axis and the curve y=cos(x^2) A)find the exact volume of the solid of revolution obtained by rotating R about the y-axis B) find the exact volume of the solid of revolution obtained by rotating R about the x-axis i am lost on the...

I have a volume problem that I has been bothering me for a while now, as I have just not been able to figure it out. The question involves finding the volume of the solid generated by revolving the region bounded by y = x, y = 0, and y = 4 around the line x = 6. I tried doing \int ^{4}_{0}16...

Homework Statement I'm having a bit of trouble when it comes to volume of revolutions and areas. I find it quite difficult when it comes to setting up the integral. I'm not sure when to use the shell or washer method. Could someone explain to me or give me a tutorial on how to set up the...

I'm having a bit of trouble when it comes to volume of revolutions and areas. I find it quite difficult when it comes to setting up the integral. Could someone explain to me or give me a tutorial on how to set up the equations thanks! Here are a few examples The region enclosed by the...

I got into an argument with someone over Earth revolving around the sun vs sun revolving around the earth. He stated that there are reference points within the universe where the sun appears to revolve around the Earth that are NOT on the surface of the earth. I conceded that perhaps a point...

Homework Statement A circle with a radius of (a/2) is bored through the centre of a sphere of radius a. Find the volume of the remaining solid. Homework Equations The Attempt at a Solution I've been trying this for an hour now and I've been trying to find the remaining volume by...

Hi, I have the area D(x,y): \sqrt{x}e^{x^2}\leq y \leq 3,~~ 0\leq x \leq 1 That is rotated about the x axis, and i need to calculate the area \pi \int_0^1 3^2-y^2 = \pi \int_0^1 9-xe^{2x^2} \frac{-9\pi}{4}\cdot (e^{2x^2}-1)\bigg|_0^1 But this is all wrong, why?

Hi, The area e^x-1 Is rotated about the y axis, bounded by y=1, x=0 and x=ln2 find the volume of the solid. And i am clearly making something wrong, so if anyone could verify my work. ~ 2\pi\int_0^{ln2}x\cdot(1-(e^x-1) -2\pi\int_0^{ln2}xe^x-2x Integration: u=x, du=1 dv=e^x, v=e^x...

Homework Statement A hole is drilled through the center of a ball of radius r, leaving a solid with a hollow cylindrical core of height h. Show that the volume of this solid is independent of the radius of the ball. Homework Equations V = \int_{a}^{b} 2\pi x (f(x) - g(x))...

Homework Statement A science fiction tale describes an artificail "planet" in the form of a band completely encircling a sun, the inhabitants living on the inside surface (where it is always noon). Imagine the sun is like our own, that the distance to the band is the same as the Earth-Sun...

Homework Statement Homework Equations integral [a, b] 2*pi*x*sqrt(1 + (dy/dx)^2) The Attempt at a Solution Well I know how to do surface area questions... But that the @#$@ is with this random equation? How would I even start to evaluate it... Like honestly... I don't even understand the...

R(x)=x^3 bounded by x=0, x=2 and y=1. a. revolved around x=2 b. revolved around x=10 my pathetic attempt: a. v=pi[(integral from 0 to 1)(2-y^1/3)^2]dy so =pi[4y-3y^(4/3)+(3/5)y^(5/3)]evaluated from 0 to 1 =(8/5)pi b. v=pi[(integral from 0 to 2)(10-x^3)^2]dy I must admit that I...

Homework Statement Find the volume of the solid that results when the region bounded by the curves y=16-x^{2} and y=16-4x is revolved around x=8. If you could show me how to do it with both the shell method and washer method it would be greatly appreciated. Homework Equations...

Homework Statement Find the volume of the solid obtained by rotating the region bounded by the curves y=0,y=sin(6x),x=6,x=0 about the axis x=−4. Homework Equations cylindrical method? circumference*thickness*height The Attempt at a Solution So I'm not sure how to write the...

Homework Statement Find the volume of the solid obtained by rotating the region in the first quadrant bounded by the hyperbola y2−x2=4 and the lines y=0, x=3 and x=5 about the y− axis. Homework Equations Nothing specific...general equations The Attempt at a Solution So I would...

Homework Statement Set up, but do not evaluate, an integral for the area of the surface obtained by rotating the curve y=xe-x 1=<x=<3 about the y-axis. Homework Equations S=integral from a to b x 2pix ds where ds=sqrt(1+(dy/dx)2)dx The Attempt at a Solution The first thing I...

Homework Statement Find the volume of the solid generated by revolving the triangular region bounded by the lines y= 2x, y= 0 and x= 1 about the line x= 1. Homework Equations V= \int A(x)dx = \int \pi[R(x)]^{2}dx The Attempt at a Solution I used the disk method, in which I found...

Homework Statement let f(x)=x^3+x^5. Evaluate int((f(x)^-1)^2, x = 0 .. 2) The Attempt at a Solution i have a feeling that it has a relation with counting volume of a solid revolution.but i don't know how to answer it...