Draft:Cross section integration

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:

$\int _{a}^{b}A(x)dx$

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:

$\int _{c}^{d}A(y)dy$

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:

$\int \limits _{a}^{b}(f(x))^{2}dx$

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:

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

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:

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

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:

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

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:

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

References

"Volumes of Solids with Known Cross Sections". CliffsNotes.com. Retrieved May 14, 2024.

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