Volumes Definition and 182 Threads

Quick question, may seem rather dumb - but I just want to make sure of something.. Question: Find the volume of the solid obtained by rotating about the y-axis the region between y = x and y = x^2 so when I am setting up my integral am I correct in saying $$TOP - BOTTOM i.e. --> \int^1_0...

Why is it that forces (for changes in pressure across a fluid) on the control volume are generally considered compressive? Even in the derivation of the Navier-Stokes equation, it is assumed that forces from fluid pressure will be compressive? Why?

Here are the questions: I have posted a link there to this thread so the OP can view my work. edit: This question has since been deleted at Yahoo! Answers.

Here is the question: I have posted a link there to this topic so the OP can see my work.

This topic is for questions and comments pertaining to this topic: http://mathhelpboards.com/math-notes-49/volumes-pyramids-6131.html

if i had 500ml of water and added 250ml of propanol, how would I go about working out its percentage by volume? thanks

Homework Statement 1) A cylinder with an initial volume of 1.5 m3 initially contains water 1 MPa and 200 ° C (condition 1). Container is cooled at constant temperature until the volume is 42% of the initial volume (state 2). The constant temperature process is followed by a constant volume...

I'm trying to get started on this project but am totally confused about how to find the volume of the solid. All the information I was given was the following: y= √x boundaries: 0,9 cross sections: isosceles right triangle how the hell do I get started?!

Homework Statement Calculate the volume of the body that is bounded by the planes: x+y-z = 0 y-z = 0 y+z = 0 x+y+z = 2 Homework Equations The Attempt at a Solution I made a variable substitution u = y+z v = y-z w = x which gave me the new boundaries u+w = 2...

Hi to everybody. I´m reading a book about statics and I cannot understand this chapter. I have been calculating moments of forces in hundreds of problems, when I found a force acting on a body I needed to fix a coordinate system, then calculating the moment arms of that force around a point...

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 Find the volume of the solid formed by revolving the functions: y=x2 and y=1 about the line y=6 Homework Equations ∏∫(ro2-ri2)dx The Attempt at a Solution I found the outer radius to be (1-x2)-6 and my inner radius to be -6. Also, the limits of integration were...

(Hi everyone! Apologising for the trivial(and likely boring) question in advance. Sadly, it has me boggled for some reason). Homework Statement In a certain temperature and under a pressure, 500 mL of H2, and 100 mL of O2 are poured into a container. A Chemical reaction occurs as follows...

Homework Statement THe region bounded by y = -x + 3 and y = x^2 - 3x the region revolve about a, x-axis, and b, y=axisHomework Equations V = π∫r^2 dxThe Attempt at a Solution I have no clue to solve it since the volume overlap. I try to ignore the overlapped region but didn't get the right...

Homework Statement Find the volume which lies below the plane z = 2x + 3y and whose base in the x - y plane is bounded by the x- and y-axes and the line x + y = 1. Homework Equations I = \int\int_{R} f(x, y) dydx = \int^{b}_{a}\int^{y=y_{2}(x)}_{y=y_{1}(x)} f(x, y) dydx The...

Hi everyone, I'm a nuclear engineering student so my chemistry knowledge is limited to a couple of courses in first year so please be patient with me. I was wondering how difficult or expensive would it be to produce hydrocarbons with deuterium instead of hydrogen on a large scale? (say...

Hi How much volume would be in the following round pipes 1. 1000mm x 50mm 2. 1000mm x 76mm 3. 1000mm x 101mm 1 liquid has a S.G = 1 1 liquid has a S.G = 1.30 Any help appreciated IJC7

Homework Statement Find the volume of the solid generated by revolving the region between the parabola x = y^2 + 1 and the line x = 3 about the line x = 3. Homework Equations The answer is found by integrating with respect to y with disk method, but I don't understand why my answer is...

The integral form of gauss's law is used to determine the electric field of charge distributions which possesses a certain amount of symmetry. Now imagine using it in situations where the gaussian surface includes equal amounts of positive and negative charge. For example,imagine a point...

Homework Statement For an assignment, I'm required to design three water bottle using 3 different polynomial functions. I've used a linear, quadratic and cubic. The first bottle needs to be 600ml, the second 300ml, and the third 1L. In order to 'create' the bottles, we are to calculate the...

Homework Statement Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer. y=x, y=0, x=2, x=4; about x=1 The Attempt at a Solution one of the radii is x=y but I am...

How do I solve this? How do I determine the range? Ill they be triple integrals?Please explain to me. Find the volumes in R3. 1. Find the volume U that is bounded by the cylinder surface x^2+y^2=1 and the plane surfaces z=2, x+z=1. 2. Find the volume W that is bounded by the cylindrical...

Hello, In R^3, the surface of the parallelogram determined by two vectors u and v is given by the norm of the cross product of u and v. For my research, I have to know if this can be generalized in the following manner: Let e_1,..,e_n be the canonical basis of R^n, and Ext_k be the exterior...

I am trying to find proofs on the internet of how cylindrical shells disks work but I can't find any. Are there any proofs out there?

Homework Statement Find the volume of this equation, revolved around x axis Homework Equations y=x^2 y^2=x The Attempt at a Solution 1) (pi)(r^2) 2) r = x^2 3) (pi)((x^2)^2) 4) (pi)(x^4) now to integrate 5) (pi)(1/5(x^5)) since x = 1, and 1^5=1, 1/5=1/5 pi/5? there...

What are the mathematical prerequisites of these books? In particular, what are the mathematical prerequisites of volume I?

Homework Statement Two thermally insulated vessels are connected by a narrow tube fitted with a valve that is initially closed as shown in the figure. One vessel of volume V1 = 15.2 L, contains oxygen at a temperature of T1 = 280 K and a pressure of P1 = 1.77 atm. The other vessel of volume V2 =...

I am currently learning about volumes of revolution in calulus, and have looked ahead to surfaces of revolution as well. I want to try and extend this concept to revolving 3d functions over the x-axis into the fourth dimension. I found this thread...

I am considering the economic viability of a project that must accurately (within a tolerance of, say, 0.5ml) dispense volumes of water from a tank at a constant volumetric flow rate of around 50ml/s. Within the tank, the water may be held at pressures between 1 and 6bar; and at temperatures...

Homework Statement A cathedral dome is designed with three semi circular supports of radius r so that each horiontal cross section is a regular hexagon. Show that the volume of the dome is r^3 * sqrt(3) an accompanying figure - http://imgur.com/3fSqh Homework Equations...

Calculate the volume of each reagent required to prepare 25.0 mL of each standard solution at the specified KSCN concentration. Fill in the rest of the table (attached). My complete guess is that the solution II needs to be incremented by 2.5 ml each time.

Is there a general rule when to use the shell or washer method when working with calculating volumes of functions rotating about an axis? For instance should I use the shell method when rotating about the y-axis and use the washer method when rotating about the x?

Homework Statement HEY EVERYONE! Sorry for putting up another quesiton, but I'm almost done my assignment for the week. I understanding volumes by cylindrical shells, but i get confused when rotating about a line that's not the x or y axis. I have uploaded the two questions. Homework...

Currently my class it calculating volumes of solids by rotating them about some axis, say for instance the function f(x) = x^2 bounded by s = { (x,y) | 0≤x≤1 , 0≤y≤1} and rotating it about the y - axis. I understand the general look of the graph on paper but I can't visualize the actual solid...

Evaluate ∫∫R 5x2 + 2y2 where R is triangle (1,1) (2,0) (2,2) I see the lines bounding the triangle are y=x y=2-x and x=2, and have tried many attempts at setting up the correct limits. Would it be correct to split this into 2 triangles, or are the limits y=x∫y=2-x for y and 2∫1...

I am reading Julian Havil’s book Nonplussed, and in one chapter he’s discussing hypercubes, he says that the volume of an n-dimensional cube of side length L is L^n; then he goes on to note that as n-> infinity, the volume goes to zero if L<1; volume goes to 1 if L=1, and volume goes to infinity...

Problem solved I'm not sure how easy this will be to understand without a diagram, but I don't know how to upload one :( Homework Statement Let a and b be constants, with a > b > 0.A torus is formed by rotating the circle (x - a)^2 + y^2 = b^2 about the y-axis. The cross-section at y =...

Homework Statement Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis. y= 4x2, y=-2x+6 Homework Equationsy= 4x2, y=-2x+6 These 2 equations meet at x= -3/2 and x=1 integral from a to b of...

Homework Statement The area under the graph of the function y = cos inverse x on the interval [0; 1] is rotated about the x-axis to form a solid of revolution. (a) Write down the volume V of the solid as a denite integral with respect to x according to the disc/slicing method. Do NOT...

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...

Homework Statement In the question your working with the region bounded by the two curves y=x^2 and y=x^3. In the first part of the question you had to revolve the region around the x-axis and find the area which I managed to do by subtracting two areas from each other. The second part of...

Most of the time I can visualise whether some solids of revolution are annulous or not but sometimes I just don't see it. Can anyone tell me if I am missing something? Is there any way of knowing if the cross section is annulous? Please help...

I was just reading a book on low-temperature physics and stumbled upon a graph that shows the energy of liquid Helium-4 in relation to its molar volume. The graph includes both zero-point energy and the potential energy of the liquid, and the latter goes steeply (basically a vertical line) from...

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 Calculate entropy change if two equal volumes of gas (1 cm3) at standard temperature and pressure are separated from each other by a partition which is removed to create twice the volume at standard temperature and pressure Homework Equations dS = k.N. dV/V + C/T dT...

Hello, Homework Statement My problem regards the disk|washer, and cylindrical shell methods for finding volumes in single variable calc. My problem is basically am I understanding these two methods and their relationships properly. Fundamentally, these methods are indentical, as we can...

Homework Statement A right circular cone of diameter r cm and height 12 cm rests on the base of right circular cylinder of radius r cm. their bases are in the same plane and the cylinder is filled with water up to a height of 12 cm. If the cone is then removed, find the height to which water...

Homework Statement The arch y= sin(x), x is in [0, pi], is revolved around the line y=c, where c is a constant in [0, 1], to generate a solid... Anyway, then I have to represent the volume of the solid as a function of c and other stuff. Homework Equations The Attempt at a Solution...

Hi, I am having a lot of trouble with this problem. Homework Statement A lumberjack is preparing to cut down a large maple tree. With a chain saw, he makes a horizontal cut exactly halfway through the trunk, and then makes a second cut at 45 degrees, meeting the first cut along the...

(Holographic principle) Volume of a sphere is proportional to R^3 However, max amount of information inside is proportional to it's surface, to R^2 So information density is proportional to 1/R It means that if we take bigger volumes, the content inside appears to be correlated with the...