Sure, I can show you the process of calculating the volume of a sphere using integration.
The formula for the volume of a sphere is V = (4/3) * π * r^3, where r is the radius of the sphere.
To calculate it using integration, we can consider the sphere as a stack of infinitesimally thin circular disks, and integrate the area of these disks.
First, let's consider a small slice of the sphere at a distance x from the center. The radius of this circular slice will be given by r(x) = √(R^2 - x^2), where R is the total radius of the sphere.
The area of this infinitesimal disk can be calculated as dA = π * r(x)^2 dx.
To find the total volume, we integrate this area over the entire range of x, which is -R to R:
V = ∫[from -R to R] dA = ∫[from -R to R] π * r(x)^2 dx.
Plugging in the expression for r(x), we have:
V = ∫[from -R to R] π * (√(R^2 - x^2))^2 dx.
Simplifying further, we get:
V = ∫[from -R to R] π * (R^2 - x^2) dx.
Now, we can integrate this expression:
V = π * ∫[from -R to R] (R^2 - x^2) dx.
V = π * [(R^2x - (x^3)/3)] [from -R to R].
Evaluating the definite integral, we have:
V = π * [(R^2 * R - (R^3)/3) - (R^2 * (-R) - ((-R)^3)/3)].
Simplifying this expression, we get:
V = π * [2R^3 - (2/3)R^3].
V = (4/3) * π * R^3.
This is the same formula we started with, so our integration method confirms the volume of a sphere!
I hope this explanation helps you understand how to calculate the volume of a sphere using integration.
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发表于 2023-6-30 15:31
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