Question

True or False? Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. The volume of the sphere \(x^{2}+y^{2}+z^{2}=1\) is given by the integral \(V=8 \int_{0}^{1} \int_{0}^{1} \sqrt{1-x^{2}-y^{2}} d x d y\).

Short answer

The statement is false. The integral does not define the correct calculation for the volume of a unit sphere. The integral only accumulates the volume for the 1st octant, hence giving a volume that is one eighth of the actual volume. When multiplied by eight, the calculated volume will match the expected volume of a unit sphere, \(\frac{4}{3}\pi\).

Step-by-step solution

Calculation of Integral

First, the value of the given integral needs to be calculated, using limits of integration from 0 to 1 for both x and y. To evaluate the double integral, compute the inner integral with respect to \(x\) first, and then the outer integral with respect to \(y\).

Comparison to Expected Volume

The result of the integral will be multiplied by 8 based on the given equation for the volume. This result is then compared with the known volume of a unit sphere, \(\frac{4}{3}\pi\). If the values are not equal, then the statement is false and that needs to be explained.

Explanation if False

If the calculated volume does not equal \(\frac{4}{3}\pi\), then the statement is false. An explanation is then provided regarding why the calculation does not match the known volume of a unit sphere. The given integral may not cover the whole volume of the sphere, for example.