1
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4answers
54 views

Find depth of a half-filled parabolic cross-section

Given a cross-section of an object that is parabolic in shape, how do you find the depth of the object when it is "half full". A full example given in an exam: A long trough whose cross-section ...
3
votes
2answers
55 views

How to determine the point at a set length along a given function (parabola)?

Given a specific function, a parabola in this instance, I can calculate the length of a segment using integrals to sum infinite right angled triangles hypotenuse lengths. My question is, can I reverse ...
1
vote
1answer
101 views

Finding the volume of a cone by integration of parabolic conic sections

I am working on a purely academic way of finding the volume of a right circular cone of height $h$ and radius $r$, (assume $h > r$), using integration of parabolic conic sections (conic sections ...
2
votes
2answers
49 views

Multivariable Calculus: Volume

Trying to figure out the following problem: Evaluate the integral $\int\int\int_EzdV$, where E lies above the paraboloid $z = x^2+y^2$ and below the plane $z=6y$. Round the result to the nearest ...
1
vote
1answer
124 views

Finding a mass of elipse

I have a task, to find a part of perimeter of an ellipse (on plane), I know it's density function. Both ellipse equation and density are given in cartesian coordinates. So I set up a line integral, ...
0
votes
2answers
1k views

Find the surface area of the part of the paraboloid $ z=5-(x^2 + y^2)$ that lies between the planes $z=0$ and $z=1$.

I have the following math question: Find the surface area of the part of the paraboloid $ z=5-(x^2 + y^2)$ that lies between the planes $z=0$ and $z=1$. So far i have computed $\sqrt{fx^2+fy^2+1}$ ...
1
vote
0answers
43 views

How can I find $\int{\sqrt{\left(b^2-1\right)x^2+1\over-x^2+1}}dx$?

I got this from the perimeter of an ellipse. I came up with the formula: arclength of f(x) for x from a to b=$\int_a^b\sqrt{f'(x)^2+1}dx$. Since an ellipse has the equation: $$\left({x-h\over ...
1
vote
1answer
657 views

Incomplete elliptic integral of the second kind and the arc length of an ellipse - does a `simple` relation exist?

Short introduction For a calculation I am working on I need to determine the arc length $l$ of a part of an ellipse in terms of the major axis $2a$, the minor axis $2b$ and the angle $\phi$. I ...
1
vote
1answer
399 views

how to calculate the double integral over the intersection of an ellipse and a circle

How to calculate the double integral of $f(x,y)$ within the intersected area? $$f(x,y)=a_0+a_1y+a_2x+a_3xy$$ $a_0$, $a_1$, $a_2$, and $a_3$ are constants. The area is the intersection of an ellipse ...
1
vote
2answers
4k views

Set up double integral of ellipse in polar coordinates?

How do you set up a double integral for an ellipse in polar coordinates without using Jacobian or Greens Theorem? I can't seem to figure out what (or if) the limits of r can possible be. $x = ...
1
vote
1answer
2k views

Moment of inertia of an ellipse in 2D

I'm trying to compute the moment of inertia of a 2D ellipse about the z axis, centered on the origin, with major/minor axes aligned to the x and y axes. My best guess was to try to compute it as: ...
3
votes
1answer
141 views

The probability of $Ax^2+Bxy+Cy^2 = 1$ defining an ellipse.

In Keith Kendig's paper, Stalking the Wild Ellipse (published in the American Mathematical Monthly, November 1995), he says that if $A, B, C$ are chosen at random, the probability that the Cartesian ...