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Prasun-rick
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Is it possible to find the volume of a sphere(i know the formula) using definite integration ? And if possible how to proceed ??
Thanks in advance
Thanks in advance
Hello Prasun-rick,Prasun-rick said:Is it possible to find the volume of a sphere(i know the formula) using definite integration ? And if possible how to proceed ??
Thanks in advance
Can you just pose the integral equation of the volume of the figure you posted !BvU said:https://www.physicsforums.com/attachments/104413
Maybe I confused you. The result of this revolution is a disk like this and that volume is relatively easy to express in a and b ...
It's not an integral. It's a disk. I lost the picture, here it is again, with the question: what is the volume, expressed in a and b ?:Prasun-rick said:Can you just pose the integral equation of the volume of the figure you posted !
Maybe pi*(r^2-x^2)*dx ??BvU said:Maybe this is better :
View attachment 104424
yeah maybe ∫pi*(r^2-x^2)dx with integral limits from -r to r ??BvU said:Perfect. Next step: we are going to add up all these disks from -r to +r and if we take the limit for ##dx\downarrow 0## we get an integral. Can you write down that integral ? (it's an easy question, because you have almost all of it already...)
ThanksBvU said:Bingo
But what is thatPrasun-rick said:Thanks
/iiint dV
Is it okay for me to learn triple integral now?? And if yes then where to start??BvU said:It means you want to sum up infinitesimal volume elements ##dV##. A volume like a sphere has three dimensions, so three integrations are necessary. In cartesian coordinates you get the volume of a cube at the origin with size ##a## from $$
\int\limits_0^a \int\limits_0^a\int\limits_0^a dxdydz = a^3$$For a sphere the limits are unwieldy in cartesian, but comfortable in spherical coordinates. A volume element is ##r^2drd\phi d\theta## (##\theta## is azimuthal) so you get $$
\int\limits_0^{2\pi} \int\limits_0^\pi \int\limits_0^R \; r^2 \sin\theta \;d\phi d\theta dr = \\
2\pi \int\limits_0^\pi \int\limits_0^R \; r^2 \sin\theta \;d\theta dr =
4\pi \int\limits_0^R \; r^2 \; dr = \ ...
$$
(There are alternative notations, like ##\ \int\limits_{\rm Volume} d^3 V\ ##)
And please teach me how to give such prominent integral sign like the ones you are typingPrasun-rick said:Is it okay for me to learn triple integral now?? And if yes then where to start??
Depends on what book or curriculum you are following; hard for me to answer. But it's always good to satisfy one's curiosity (is my opinion).Prasun-rick said:Is it okay for me to learn triple integral now?? And if yes then where to start??
The formula for finding the volume of a sphere using integration is V = ∫∫∫ dV = ∫∫∫ r^2 sin(φ) dr dθ dφ, where r is the radius of the sphere and φ and θ are the angles that define the position of a point on the surface of the sphere.
Integration is used to find the volume of a sphere because it allows us to calculate the volume of a three-dimensional object by summing up an infinite number of infinitesimal volumes. In the case of a sphere, integration allows us to sum up the volume of an infinite number of thin slices that make up the sphere, resulting in the total volume.
The limits of integration for finding the volume of a sphere are 0 to 2π for θ, 0 to π for φ, and 0 to r for r. This is because we need to integrate over the entire surface of the sphere in all three dimensions.
Yes, the volume of a sphere can also be found using the formula V = 4/3πr^3, where r is the radius of the sphere. This formula is derived from the geometric properties of a sphere and does not involve integration.
The volume of a sphere using integration is related to its surface area through the formula V = ∫∫∫ dV = ∫∫∫ r^2 sin(φ) dr dθ dφ, which can also be written as V = 4π∫∫ r^2 sin(φ) dr dθ. This shows that the volume of a sphere is directly proportional to its surface area, with a constant factor of 4π.