Revolution Definition and 393 Threads

need some help here. 1. find the volume (by washer method) of the solid generated by revolving the region bounded by $y=3+x^2$ and the line $y=4$ about the x-axis. 2. write the integral that will give the volume of the solid generated by revolving the region bounded by $y=e^{2x}$, $x=-1$ and...

Despite the 20+ answers in this thread I think most of us would agree that life for 99% of fantasy world inhabitants would be awful. Strip away the royal courts, the fellowships, the magic schools etc and most of the time you're left with a feudalistic world full of peasants that have to work 10...

Hello MHB, As students of calculus, we are taught to find the volumes of solids of rotation obtained by revolving given regions about horizontal and vertical axes of rotation. But, what if the axis of rotation is neither horizontal nor vertical? Please consider the following diagram: We wish...

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A solid lies between planes perpendicular to the x-axis at x=-a and x=a for values of a>0 to be given below in parts (i) and (ii). In each case the cross-sections perpendicular to the x-axis between these planes run from the semicircle y=√(a^2-x^2) to the semicircle y=-√(a^2-x^2). If...

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Homework Statement Let R be the region between y=tan(x) and the x-axis from x=0 to x=pi/2. Find the volume of the solid formed when R is revolved around the y-axis.Homework Equations Please try to solve this problem using elementary calculus. The textbook is an elementary calculus textbook...

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find the volume of the solid generated by the revolution of the curve $y^2 (2 a - x) = x^3$ about its asymptote.

Homework Statement A planet of mass ##M## moves around the Sun along an ellipse so that its minimum distance from the Sun is equal to ##r## and the maximum distance is ##R##. Making use of Kepler's laws, find its period of revolution. (Ans: ##\pi \sqrt{(r+R)^3/(2GM)}##) Homework...

Homework Statement Find the volume of solid obtained by rotating the region bounded by y=x^2,y=4 and x=0 about the line x=-2 using the shell method. Homework Equations I'm stuck because using disks I got the right answer 136pi/3 but I can't using shells? The Attempt at a Solution...

Hey guys I'm stuck on this problem. Its an easy one but I need some help.. It's asking for the volume of the solid obtained by rotating the region bounded by y=x^2, y=4, x=0 about the line x= -2 using the shell method. I got the answer correct using the disk method (answer is 136 pi/3)...

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Homework Statement What multiple of the distance between the centre of the Earth and that of the moon is the distance between the centre of the Earth and a geostationary satellite which above a fixed location on the equator?Take the cycle of revolution to the moon around the Earth to be 27...

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Hi, this isn't a specific question but say you had a function y = x - 1 and you were told that the region from x = 1 to x = 3 was rotated 2pi radians and were asked to find the volume of revolution formed. My question is, would this volume of revolution be the same if they said it was rotated...

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Homework Statement Find the volume of the solid generated by revolving about the line x=-1, the region bounded by the curves y=-x^2 + 4x - 3, and y=0. Homework Equations Shell Method? The Attempt at a Solution V= 2pi * ∫x* f(x) dx, where a and b are the lower and upper limits of...

Homework Statement Find the volume of the solid rotated about the given axis Homework Equations ∫R^2-r^2dx Disk method The Attempt at a Solution I'm having trouble finding the limits of integration: here's my setup Pi∫(x^3+1)dx and integrated from -1 to 1. I got this by setting...

A couple of quick questions after watching a video on the helectic model that the solar system follows on it's course around the galactic center. Please bare with me these maybe idiotic questions. A) Are black holes bound to the spin of the galaxy or do they sit in place on the galactic plane...

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How come the area below the graph of $$\frac 1x$$ between $$[1, \infty)$$ does not exist, but the solid of revolution below the same graph in that same interval does exist? I do not see the logic.

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Homework Statement The region R shown in Fig. 1 is bounded by the line y=8(x-2), the axes, and the line y=h. Find by integration the volume formed when R is rotated through 360° about the y axis(see Fig. 2) A whisky glass has the shape indicated in Fig 2 where the units are centimetres...

Homework Statement In a certain binary-star system, each star has the same mass which is 7.5 times of that of the Sun, and they revolve about their center of mass. The distance between them is the 7.5 times the distance between Earth and the Sun. What is their period of revolution in years...

Calculate the volumes of the rotation bodies which arises when the area D in the xy-plane bounded by x-axis and curve $$7x-x^2$$may rotate around x- respective y-axes. I will calculate $$V_x$$ and $$V_y$$ I start to get crit point $$x_1=0$$ and $$x_2=7$$ rotate on y-axe: $$2\pi\int_a^bf(x)dx$$...

Homework Statement Calculate the volume of the solid of revolution formed by rotating the region around the y-axis. Apply the shell method. f(x)=e^x, x=0, y=8 Homework Equations V=∫2∏x((f(x))-g(x))dx The Attempt at a Solution This is what I did: (I integrated from 0 to 8)...

Earth is revolving around the Sun at a great speed. Our Galaxy is also moving away from other galaxies at a great speed and so on and so forth there are many kinds of motions that we are subjected to ( Earth's rotation, galaxy's rotation etc.). Plz tell 1. Are all these motions uniform (steady)...

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Homework Statement Find the volume of 2sin(x) and -sin(x) from 0 to pi revolving around the y-axisThe Attempt at a Solution My problem is with the geometry of this problem, 2sin(x) is above the x-axis and -sin(x) is below the x-axis. My belief was that I should be adding on the extra area of...

Homework Statement A torus is formed by revolving the region bounded by the circle x2 + y2 = 1 about the line x = 2. Find the volume of the solid. The Attempt at a Solution I've actually got the answer correct. I'm using shell method. My height, h(x), I believe is double the area of...

Homework Statement Find the volumes of the solids generated a) x-3y+3= 0, x=0 , y=2 about the x axis I sketched the graph got a straight line, I then proceeded to integrate y2 ∏∫y2δx =[(x^3/27)+(x^2/3)+ (x)] from x=3 to x=-3 I got 8∏ but the answer is 5∏ b) x-y2-1=0, x=2...

Homework Statement Find the volume of the solid generated by rotating about the y-axis y= 1-x3 x=0 , y=0 I tried sketching the graph of y= 1-x^3 then tried to find the volume from y=1 to y=0. if x^3 = 1-y x= (1-y)1/3 so x2= (1-y)2/3 ∏∫x2δy => ∏∫(1-y)2/3.dy = ∏[ -3/5(1-y)^(5/3)] I did that...

Homework Statement Let n>1/2 and consider the function f(x)=x^{-n} for x\in[1,∞) Calculate the volume of the solid generated by rorating f(x) about the x-axis, showing all details of your working. Homework Equations Since it is rotated about the x-axis, its axis of symmetry is...

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Homework Statement (a) Consider a binary star system in which the two stars have masses M1 and M2 and the stars move on circular orbits separated by a distance R. Derive the formula for the period of revolution. (b) Suppose M1= 1.22M and M2= 0.64M (where M = mass of the sun) and R= 0.63...

1. Use the integration capabilities of a graphing utility to approximate the surface area of the solid of revolution. (Round your answer to four decimal places.) Function: y = sin(x) Interval: [0, pi/4] revolved about the x-axis 2. Use the area of a surface of revolution...

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Is it possible to revolve a function around y = x? If so how would you do it? I suppose the main difficulty is in finding the radius for the area of a disk or cylinder. Is there any method that works will all or most functions?

Homework Statement Volume of the region bounded by y = x^2 and x = y^2 about y = 1 Homework Equations \pi r^2 The Attempt at a Solution So the functions look something like this: I decided to use method of washers with respect to x. The radius if the center is at y = 1 of...

Homework Statement A toy truck moves around the outside of a circle of radius 0.6m at 2 revolutions per second Calculate: a: the angular speed of the truck b: the linear speed of the truck c: the period of revolutionHomework Equations ω= \frac{2π}{T} \\ v=rω ?? The Attempt at a Solution For...

Suppose I have a region R whose boundary extremely complicated. While it would take me hundreds of years to approximate the boundary with formulae, I can easily estimate the area of R within a desired precision. I want to find the volume of the solid of revolution of R . My intuition told...

Here is the question: Here is a link to the original question: Find the volume V of the solid obtained by rotating the region bounded by x=16-(y-3)^2, x+y=7 about the x-axis? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response. First, let's take a look...

Homework Statement From K&K's 'Intro to Mechanics' Find the shortest possible period of revolution of two identical gravitating solid spheres which are in circular orbit in free space about a point midway between them. Homework Equations The Attempt at a Solution So I figured...

Homework Statement Planet A (at X meters) completes 1 full rotation (Y sec). Planet A then shrinks to (X2 meters) What is its rotation speed now. Conservation of angular momentum Homework Equations T=2pi/w W=V/r The Attempt at a Solution I found how to reverse engineer the...