Volumes with Cross Sections: Squares and Rectangles

Abigail Young
8 min read
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Study Guide Overview
This study guide covers calculating volumes of solids with known cross-sections using integration. It focuses on solids with square and rectangular cross-sections. Examples demonstrate how to set up the integral, determine the bounds, and calculate the volume using representative cross-sections perpendicular to the x-axis and y-axis. The key formula used is V = ∫[a,b] A(x) dx , where A(x) represents the cross-sectional area.
#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:
where is a function for the area of a cross-section (some two-dimensional shape) perpendicular to the x-axis on the closed interval and represents its thickness.
#🔲 Square Cross Sections
To find a shape using square cross sections, we’ll use the formula for . Plugging this into our initial equation, we find that the formula for the volume of a solid with square cross sections is
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 .
#🔳 Rectangular Cross Sections
To find the area of a rectangle, we use the formula where is the width of the rectangle and is the height. So the formula for the volume of a soil with rectangular cross sections is given by . Remember, is your thickness!
#✏️ Solving Cross-Section Problems
🤔 Now that we have these formulas, how do we figure out what , , or 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 and forms the base of a solid and each cross section perpendicular to the 𝑥-axis is a square. What is the vo...

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