Volumes with Cross Sections: Squares and Rectangles

#8.7 Volumes with Cross Sections: Squares and Rectangles

So far in this unit, you’ve been learning how to find the area between two curves. However, we can also use these curves to represent a three-dimensional object. In this guide, you’ll learn how to find the volume of this object.

#🪩 Solids with Cross Sections

When we want to find the volume of a three-dimensional object, particularly those difficult to calculate with geometry, we can instead break it up into infinitely thin slices that are easier to work with. 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.

#🔲 Square Cross Sections

To find a shape using square cross sections, we’ll use the formula $s^2$ for $A(x)$. Plugging this into our initial equation, we find that the formula for the volume of a solid with square cross sections is

$$V = \int_a^b s^2\ dx$$

It’s important to note here that we’re taking the volume of rectangular prisms—it’s just that the thickness is infinitely thin, represented by $dx$.

#🔳 Rectangular Cross Sections

To find the area of a rectangle, we use the formula $w*h$ where $w$ is the width of the rectangle and $h$ is the height. So the formula for the volume of a soil with rectangular cross sections is given by $V = \int_a^b w * h \space dx$. Remember, $dx$ is your thickness!

#✏️ Solving Cross-Section Problems

🤔 Now that we have these formulas, how do we figure out what $s$, $w$, or $h$ are so that we can use them? Let’s work through an example.

#Example 1: Solids with Square Cross Sections

Suppose a region bounded by $y = x^2$ and $y = \sqrt{x}$ forms the base of a solid and each cross section perpendicular to the 𝑥-axis is a square. What is the vo...