6 min read•february 15, 2024
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!
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. In this guide, we’ll explore this same concept using different shapes.
The formula for the area of a triangle depends on what kind of triangle it is. This means that the function 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.
The formula for the area of an equilateral triangle is given by the equation where 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 .
The formula for the area of a right-angled isosceles triangle is were 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 .
To find the area of a circle, we can use the familiar formula , where is the radius of the circle. A semicircle only has half the area of a circle though, so we’ll use the equation for A(x) in our volume formula. That means that for a solid with semicircular cross sections, its volume will be found using the equation .
Now that we have our area functions we need to find or in order to use them! Let’s work though an example.
Suppose a region bounded by and forms the base of a solid with cross sections taken perpendicular to the x-axis. What integral will give the volume of the solid if the cross sections are equilateral triangles? What about semicircles?
Question courtesy of Flipped Math
🧐We’ll begin by drawing a picture of out graphs so we can visualize the problem and identify the bounds we’ll use when evaluating our integral.
In this image, forms the upper curve, and is the bottom curve. Let’s let function and . The area in grey is the base of our solid. The purple line represents a singular cross section of this solid. You can imagine that the cross section is coming towards you along a z-axis not shown in this drawing. If we rotated our graph and included the z-axis, our graph would look like the one on the top for equilateral triangle cross sections and the one on the bottom for semicircular cross sections.
👀Graphically, we can see that the two functions intersect for and . This means that our bounds are given by the closed interval [0, 1]. We could also find the bounds algebraically by setting the two functions equal and solving for x as shown below.
You’ll notice that one side of the equilateral triangle at a point is given by . So, the side length of the equilateral triangle cross section is given by .
Now that we have a value for , we can identify our functions for this solid. For the equilateral triangle, this will be given by .
🪢 Let’s tie it all together now and find our volume integrals. Plugging in our bounds and functions into the formula , we find that the volume of the solid with equilateral triangle cross sections is .
The diameter of the semicircle at a point is defined by . Recall that the radius of a circle is one half its diameter. Therefore, the radius of the semicircle for some is .
Now that we have a value for , we can identify our functions for this solid. For the semicircle, this will be given by .
🪢 Let’s tie it all together now and find our volume integrals. Plugging in our bounds and functions into the formula , we find that the volume of the solid with semicircular cross sections is given by .
When finding the volume of a solid with a known cross section, we can use the equation
is a function for the area of a cross section perpendicular to the x-axis on the interval [a, b].
For equilateral triangle cross sections where is the length of one of the sides of the triangle.
For right isosceles triangle cross sections were is the length of the two matching sides of the triangle.
For semicircular cross sections where is the radius of the semicircle at .
Using methods for finding the areas between curves we can determine equations to describe or and find bounds for our integral.
Plugging in our bounds and the function results in the volume integral for the solid. Evaluating this definite integral will give a value for the volume of the solid.
Axis of Revolution
: The axis of revolution is an imaginary line around which a two-dimensional shape rotates to create a three-dimensional object. It serves as the central point for rotation.Cross Sections
: Cross sections refer to the 2D shapes that are obtained when a solid is cut by a plane. They help us understand the shape and properties of the solid.Definite Integral
: A definite integral is a mathematical tool used to calculate the exact area between a function and the x-axis over a specific interval.Function
: A relationship between two sets where each input (domain) value corresponds to exactly one output (range) value.Integration
: Integration is an operation in calculus that finds antiderivatives, or reverse derivatives, allowing us to calculate areas, volumes, and other quantities related to curves and functions.Limits of Integration
: The limits of integration are the values that determine the range over which an integral is evaluated. They specify the starting and ending points on the x-axis for finding the area or calculating other quantities using integration.Radius
: The radius is the distance from the center of a circle to any point on its circumference.Semicircular Cross Sections
: Semicircular cross sections are used to find the volume of a solid by using semicircles as the base for each cross section.Triangular Cross Sections
: Triangular cross sections refer to the method of finding the volume of a solid by using triangles as the base of each cross section.y = √(r^2 - x^2)
: This equation represents part of a circle with radius r centered at the origin (0, 0). It gives the y-coordinate of a point on the circle's circumference based on its x-coordinate.y = f(x)
: This notation represents the relationship between the dependent variable y and the independent variable x in a function. It indicates that y is determined by some rule or equation involving x.6 min read•february 15, 2024
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!
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. In this guide, we’ll explore this same concept using different shapes.
The formula for the area of a triangle depends on what kind of triangle it is. This means that the function 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.
The formula for the area of an equilateral triangle is given by the equation where 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 .
The formula for the area of a right-angled isosceles triangle is were 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 .
To find the area of a circle, we can use the familiar formula , where is the radius of the circle. A semicircle only has half the area of a circle though, so we’ll use the equation for A(x) in our volume formula. That means that for a solid with semicircular cross sections, its volume will be found using the equation .
Now that we have our area functions we need to find or in order to use them! Let’s work though an example.
Suppose a region bounded by and forms the base of a solid with cross sections taken perpendicular to the x-axis. What integral will give the volume of the solid if the cross sections are equilateral triangles? What about semicircles?
Question courtesy of Flipped Math
🧐We’ll begin by drawing a picture of out graphs so we can visualize the problem and identify the bounds we’ll use when evaluating our integral.
In this image, forms the upper curve, and is the bottom curve. Let’s let function and . The area in grey is the base of our solid. The purple line represents a singular cross section of this solid. You can imagine that the cross section is coming towards you along a z-axis not shown in this drawing. If we rotated our graph and included the z-axis, our graph would look like the one on the top for equilateral triangle cross sections and the one on the bottom for semicircular cross sections.
👀Graphically, we can see that the two functions intersect for and . This means that our bounds are given by the closed interval [0, 1]. We could also find the bounds algebraically by setting the two functions equal and solving for x as shown below.
You’ll notice that one side of the equilateral triangle at a point is given by . So, the side length of the equilateral triangle cross section is given by .
Now that we have a value for , we can identify our functions for this solid. For the equilateral triangle, this will be given by .
🪢 Let’s tie it all together now and find our volume integrals. Plugging in our bounds and functions into the formula , we find that the volume of the solid with equilateral triangle cross sections is .
The diameter of the semicircle at a point is defined by . Recall that the radius of a circle is one half its diameter. Therefore, the radius of the semicircle for some is .
Now that we have a value for , we can identify our functions for this solid. For the semicircle, this will be given by .
🪢 Let’s tie it all together now and find our volume integrals. Plugging in our bounds and functions into the formula , we find that the volume of the solid with semicircular cross sections is given by .
When finding the volume of a solid with a known cross section, we can use the equation
is a function for the area of a cross section perpendicular to the x-axis on the interval [a, b].
For equilateral triangle cross sections where is the length of one of the sides of the triangle.
For right isosceles triangle cross sections were is the length of the two matching sides of the triangle.
For semicircular cross sections where is the radius of the semicircle at .
Using methods for finding the areas between curves we can determine equations to describe or and find bounds for our integral.
Plugging in our bounds and the function results in the volume integral for the solid. Evaluating this definite integral will give a value for the volume of the solid.
Axis of Revolution
: The axis of revolution is an imaginary line around which a two-dimensional shape rotates to create a three-dimensional object. It serves as the central point for rotation.Cross Sections
: Cross sections refer to the 2D shapes that are obtained when a solid is cut by a plane. They help us understand the shape and properties of the solid.Definite Integral
: A definite integral is a mathematical tool used to calculate the exact area between a function and the x-axis over a specific interval.Function
: A relationship between two sets where each input (domain) value corresponds to exactly one output (range) value.Integration
: Integration is an operation in calculus that finds antiderivatives, or reverse derivatives, allowing us to calculate areas, volumes, and other quantities related to curves and functions.Limits of Integration
: The limits of integration are the values that determine the range over which an integral is evaluated. They specify the starting and ending points on the x-axis for finding the area or calculating other quantities using integration.Radius
: The radius is the distance from the center of a circle to any point on its circumference.Semicircular Cross Sections
: Semicircular cross sections are used to find the volume of a solid by using semicircles as the base for each cross section.Triangular Cross Sections
: Triangular cross sections refer to the method of finding the volume of a solid by using triangles as the base of each cross section.y = √(r^2 - x^2)
: This equation represents part of a circle with radius r centered at the origin (0, 0). It gives the y-coordinate of a point on the circle's circumference based on its x-coordinate.y = f(x)
: This notation represents the relationship between the dependent variable y and the independent variable x in a function. It indicates that y is determined by some rule or equation involving x.© 2024 Fiveable Inc. All rights reserved.
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