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:

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

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:

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

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:

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

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:

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

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:

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