Write a definite integral in terms of x that represents the volume of a sphere with radius 2.
since x^2+y^2+z^2 = 4
using symmetry, we can take 8 times the volume in the first octant. Then v =
8∫[0,2] ∫[0,√(4-x^2)] ∫[0,√(4-x^2-y^2)] dz dy dx
To find the volume of a sphere with radius 2 using a definite integral, we can use the method of cylindrical shells.
First, let's consider a thin cylindrical shell that is infinitesimally small in height, with radius r and thickness Δr. The volume of this cylindrical shell can be approximated by the product of its height (which is the circumference of the sphere at that height) and its surface area.
The height of the cylindrical shell is given by 2πr, and the surface area of the shell is given by 2πrΔr. Therefore, the volume of the shell is approximately 2πr * 2πrΔr.
To find the total volume of the sphere, we integrate over the range of r from 0 to the radius of the sphere (which is 2). So, the definite integral in terms of x (we will use x instead of r for integration) that represents the volume of the sphere is:
∫[0, 2] 2πx * 2πx dx
This integral represents the sum of all the cylindrical shells of varying radii, which together form the volume of the sphere.
Evaluating this integral will give us the exact volume of the sphere with radius 2.