Draft:Cross section integration

Template:AfC submission Cross section integration is a method of calculating the volumes of solids with known cross sections when integrating perpendicular to the X or Y-axis.

Definition

For cross sections taken perpendicular to the x-axis, if A(x) is a function which describes the area of a cross section of a solid on the interval [a, b], the formula for the volume of the solid will be:

$\underset{a}{\overset{b}{\int }}A(x)dx$

For cross sections taken perpendicular to the y-axis, if A(y) is a function which describes the area of a cross section of a solid on the interval [a, b], the formula for the volume of the solid will be:

$\underset{c}{\overset{d}{\int }}A(y)dy$

Specific cross sections

Square

If the cross section is a square, with its area dependent on f(x) on the interval [a, b]. The formula for the volume of the solid will be:

$\underset{a}{\overset{b}{\int }}(f(x){)}^{2}dx$

Semicircular

If the cross section is a semicircle, with its area dependent on f(x) on the interval [a, b]. The formula for the volume of the solid will be:

$\frac{\pi }{8}\underset{a}{\overset{b}{\int }}(f(x){)}^{2}dx$

Equilateral triangle

If the cross section is an equilateral triangle, with its area dependent on f(x) on the interval [a, b]. The formula for the volume of the solid will be:

$\frac{\sqrt{3}}{4}\underset{a}{\overset{b}{\int }}(f(x){)}^{2}dx$

Right triangle

Hypotenuse as base

If the cross section is a right triangle, with its area dependent on f(x) on the interval [a, b] and the hypotenuse as the base. The formula for the volume of the solid will be:

$\frac{1}{4}\underset{a}{\overset{b}{\int }}(f(x){)}^{2}dx$

Leg as base

If the cross section is a right triangle, with its area dependent on f(x) on the interval [a, b] and the hypotenuse as the base. The formula for the volume of the solid will be:

$\frac{1}{2}\underset{a}{\overset{b}{\int }}(f(x){)}^{2}dx$

References

"Volumes of Solids with Known Cross Sections". CliffsNotes.com. Retrieved May 14, 2024. Template:Drafts moved from mainspace

Larson, Ron, and Edwards, Bruce H.. Calculus of a Single Variable. United States, Brooks/Cole, 2010.