Multivariable Calculus

Multivariable calculus is the type of calculus we use to make inquiry into functions or applications involving two or more variables. For example, in statistics, it is possible to formulate the concept of a probability density function $f$ of jointly continuous random variables $X\text{and}Y$, in such a way that the volume under the surface $z=f(x,y)$ over a region $R$ in the $x$–$y$ plane is the probability that a point $(x,y)$ lies in $R$. It is important, though, in this probability application of multivariable calculus for the (density) function $f(x,y)$ to be non-negative.

Example:

Find the volume inside the ellipsoid$$\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$$

Solution: This is equal to twice the volume inside the upper half $(z\ge 0)$ of the ellipsoid. Let $V$ denote the volume inside the upper half of the ellipsoid. This volume is the same as the volume enclosed by the surface $z=c\sqrt{1-\frac{x^2}{a^2}-\frac{y^2}{b^2}}$ above the region $R$ in the $x$–$y$ plane given by$$R=\{(x,y):\frac{x^2}{a^2}+\frac{y^2}{b^2}\le 1\}=\{(x,y):-a\le x\le a,-b\sqrt{1-\frac{x^2}{a^2}}\le y\le b\sqrt{1-\frac{x^2}{a^2}}\}$$So, with $t:=b\sqrt{1-\frac{x^2}{a^2}}$, we have that $V={\int }_{-a}^a{\int }_{-t}^{t}zdydx$. Using trigonometric substitution, with $\alpha :=b\sqrt{1-\frac{x^2}{a^2}}$, set $y=\alpha \text{sin}\theta$. Then, $dy=\alpha \text{cos}\theta$ and $z=c\sqrt{1-\frac{x^2}{a^2}-\frac{y^2}{b^2}}=c\sqrt{1-\frac{x^2}{a^2}}\text{cos}\theta$, so that $zdy=bc(1-\frac{x^2}{a^2}){\text{cos}}^2\theta$, where ${\int }_{-t}^{t}zdy={\int }_{-\pi /2}^{\pi /2}bc(1-\frac{x^2}{a^2}){\text{cos}}^2\theta d\theta =bc(1-\frac{x^2}{a^2}){\int }_{-\pi /2}^{\pi /2}{\text{cos}}^2\theta d\theta =bc\frac{\pi }{2}(1-\frac{x^2}{a^2})$ Then, $V={\int }_{-a}^{a}bc\frac{\pi }{2}(1-\frac{x^2}{a^2})dx=bc\pi {\int }_0^a(1-\frac{x^2}{a^2})dx=bc\pi \cdot \frac{2a}{3}$

So, the required volume is \qquad2V=\frac{4\pi}{3}abc