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Volumes with Cross Sections: Squares and Rectangles

Abigail Young

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:

V=abA(x) dxV = \int_a^b A(x)\ dx

where y=A(x)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][a,b] and dxdx represents its thickness.

🔲 Square Cross Sections

To find a shape using square cross sections, we’ll use the formula s2s^2 for A(x)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=abs2 dxV = \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 dxdx.

🔳 Rectangular Cross Sections

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


✏️ Solving Cross-Section Problems

🤔 Now that we have these formulas, how do we figure out what ss, ww, or hh 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=x2y = x^2 and y=xy = \sqrt{x} forms the base of a solid and each cross section perpendicular to the 𝑥-axis is a square. What is the vo...

Question 1 of 9

The volume of a solid with known cross-sections is found using the formula V=abA(x) dxV = \int_a^b A(x) \ dx. What does A(x)A(x) represent in this formula? 🤔

The area of the base of the solid

The perimeter of a cross-section

The area of a cross-section perpendicular to the x-axis

The volume of a single slice of the solid