Integrating an Improper Divergent Integral & Ellipsoid Volume

[chjopl]

I need help with two questions.
Find a divergent improper integral whose value is neither infinity nor -infinity.


2. Find the volume of an ellipsoid (a^2*x^2) + (b^2*8y^2) + (c^2*z^2) = a^2*b^2*c^2 using integration.

 

[dextercioby]

For the first part,think about the (circular) trigonometrical functions and the fact that they don't have limit when evaluated in the limit of +/- infty...

Why does that ellipsoid have the that equation...??

The way i know it...

$$\frac{(x-x_{0})^{2}}{a^{2}}+\frac{(y-y_{0})^{2}}{b^{2}}+\frac{(z-z_{0})^{2}}{c^{2}}=1$$

Multiply by the square of the semiaxis' product and see whether the new expression resembles the one you're written.

Daniel.

 

[wais]

1. To find a divergent improper integral whose value is neither infinity nor -infinity, we can look at the integral of 1/x from 0 to 1. This integral is improper because it is not defined at x=0. However, when we evaluate the integral, we get ln(1)-ln(0) which is equal to 0. So the value of this divergent improper integral is 0, which is neither infinity nor -infinity.

2. To find the volume of an ellipsoid using integration, we can use the triple integral. The formula for the volume of an ellipsoid is V = (4/3)*π*a*b*c, where a, b, and c are the semi-axes of the ellipsoid. Using the given equation of the ellipsoid, we can set up the triple integral as follows:

V = ∫∫∫ (a^2*x^2) + (b^2*y^2) + (c^2*z^2) dx dy dz

We need to evaluate this integral over the entire volume of the ellipsoid, which is defined by the limits of x, y, and z. We can use the substitution method to simplify the integral. Let u = x/a, v = y/b, and w = z/c. This will change the limits of integration to -1 to 1 for all variables.

V = ∫∫∫ (a^2*u^2) + (b^2*v^2) + (c^2*w^2) a*b*c du dv dw

= a*b*c * ∫∫∫ (a^2*u^2) + (b^2*v^2) + (c^2*w^2) du dv dw

= a*b*c * ∫ (-1 to 1) ∫ (-1 to 1) ∫ (-1 to 1) (a^2*u^2) + (b^2*v^2) + (c^2*w^2) du dv dw

= a*b*c * ∫ (-1 to 1) ∫ (-1 to 1) [(a^2*u^2) + (b^2*v^2) + (c^2*w^2)] dv dw

= a*b*c * ∫ (-1 to 1) [2*a^2*u^2 +