Volumes with Cross Sections: Triangles and Semicircles

#8.8 Volumes with Cross Sections: Triangles and Semicircles

Recall from the last guide that we can instead break up 3D objects bound by curves into infinitely thin slices that are easier to work with. In this guide, we’ll apply this concept to triangles and semicircles!

#🪩 Solids with Cross Sections: Review

To find the volume of a solid with known cross-sections we can use the formula

$$V = \int_a^b A(x)\ dx$$

where $y=A(x)$ is a function for the area of a cross-section (some two-dimensional shape) perpendicular to the x-axis on the closed interval $[a,b]$ and $dx$ represents its thickness. In this guide, we’ll explore this same concept using different shapes.

#🔺Triangular Cross Sections

The formula for the area of a triangle depends on what kind of triangle it is. This means that the function $A(x)$ for a solid with triangular cross sections will be different depending on the kind of triangle that makes up the cross sections of the solid. We will develop volume formulas for solids with equilateral and right-angled isosceles triangle cross sections.

#🔻 Equilateral Triangles

The formula for the area of an equilateral triangle is given by the equation $\frac{\sqrt{3}}4s^2$ where $s$ is the length of one of the sides of the triangle. This means that the formula for the volume of a solid with equilateral triangle cross sections is given by $V = \int_a^b \frac{\sqrt{3}}4s^2 dx$.

#📐 Right Isosceles Triangles

The formula for the area of a right-angled isosceles triangle is $\frac{1}2s^2$ were $s$ is the length of the two matching sides of the triangle. So for a solid with right isosceles triangle cross sections, its volume will be given by formula $V = \int_a^b \frac{1}2s^2 dx$.

#🚫 Semicircular Cross Sections

To find the area of a circle, we can use the familiar formula $\pi r^2$, where $r$ is the radius of the circle. A semicircle only has half the area of a...