Difference between revisions of "Volume"

Latest revision as of 19:24, 17 August 2023

The volume of an object is a measure of the amount of space that it occupies. Note that volume only applies to three-dimensional figures.

Finding Volume

This section covers the methods to find volumes of common Euclidean objects.

Prism

The volume of a prism of height $h$ and base of area $b$ is $b\cdot h$ .

Pyramid

The volume of a pyramid of height $h$ and base of area $b$ is $\frac{bh}{3}$ .

Sphere

The volume of a sphere of radius $r$ is $\frac 43 r^3\pi$ .

Cylinder

The volume of a cylinder of height $h$ and radius $r$ is $\pi r^2h$ . (Note that this is just a special case of the formula for a prism.)

Cone

The volume of a cone of height $h$ and radius $r$ is $\frac{\pi r^2h}{3}$ . (Note that this is just a special case of the formula for a pyramid.)

Parallelepiped

The volume of a parallelepiped spanned by vectors $\bold{a} = a_1\bold{i} + a_2\bold{j} + a_3\bold{k}, \bold{b} = b_1\bold{i} + b_2\bold{j} + b_3\bold{k}, \bold{c} = c_1\bold{i} + c_2\bold{j} + c_3\bold{k}$ is $|\text{det}(\begin{bmatrix} c_1 & c_2 & c_3 \\ a_1 & a_2 & a_3\\ b_1 & b_2 & b_3 \\ \end{bmatrix})|).$

Irregular objects

The volume of an object defined by an upper bound of $f(x,y,z)$ in the Cartesian three-space can be found using a triple integral: $\int_{a_z}^{b_z}\int_{a_y}^{b_y}\int_{a_x}^{b_x}f(x,y,z)\text{ dx dy dz}$ , where $(a_z,b_z)$ are the bounds of $z$ and similar bounds are defined for $x$ and $y$ .

Problems

Introductory

Centers of adjacent faces of a unit cube are joined to form a regular octahedron. What is the volume of this octahedron? (Source)

Intermediate

A tripod has three legs each of length $5$ feet. When the tripod is set up, the angle between any pair of legs is equal to the angle between any other pair, and the top of the tripod is $4$ feet from the ground. In setting up the tripod, the lower 1 foot of one leg breaks off. Let $h$ be the height in feet of the top of the tripod from the ground when the broken tripod is set up. Then $h$ can be written in the form $\frac m{\sqrt{n}},$ where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. Find $\lfloor m+\sqrt{n}\rfloor.$ (The notation $\lfloor x\rfloor$ denotes the greatest integer that is less than or equal to $x.$ ) (Source)

@@ Line 14: / Line 14: @@
 The volume of a [[cone]] of height <math>h</math> and radius <math>r</math> is <math>\frac{\pi r^2h}{3}</math>.  (Note that this is just a special case of the formula for a pyramid.)
+===Parallelepiped===
+The volume of a [[parallelepiped]] spanned by vectors <math>\bold{a} = a_1\bold{i} + a_2\bold{j} + a_3\bold{k}, \bold{b} = b_1\bold{i} + b_2\bold{j} + b_3\bold{k}, \bold{c} = c_1\bold{i} + c_2\bold{j} + c_3\bold{k}</math> is <math>|\text{det}(\begin{bmatrix}
+c_1 & c_2 & c_3 \\
+a_1 & a_2 & a_3\\
+b_1 & b_2 & b_3 \\
+\end{bmatrix})|).</math>
 ===Irregular objects===
-The volume of an object defined by <math>f(x,y,z)</math> in the Cartesian three-space can be found using a triple [[integral]]: <math>\int_{a_z}^{b_z}\int_{a_y}^{b_y}\int_{a_x}^{b_x}f(x,y,z)\text{ dx dy dz}</math>, where <math>(a_z,b_z)</math> are the bounds of <math>z</math> and similar bounds are defined for <math>x</math> and <math>y</math>.
+The volume of an object defined by an upper bound of <math>f(x,y,z)</math> in the Cartesian three-space can be found using a triple [[integral]]: <math>\int_{a_z}^{b_z}\int_{a_y}^{b_y}\int_{a_x}^{b_x}f(x,y,z)\text{ dx dy dz}</math>, where <math>(a_z,b_z)</math> are the bounds of <math>z</math> and similar bounds are defined for <math>x</math> and <math>y</math>.
 == Problems ==

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