Finding the Lengths of the Bases

Oct 13, 2025 by windailys.com 93 views

Introduction

In this article, we will explore the concept of finding the volume of a solid with trapezoidal cross-sections. The solid in question has a base bounded by the curves $x = y^2$ and $x = 4$ , and the cross-sections are taken perpendicular to the $y$ -axis. The height of each cross-section is given as $h = 2$ , and the upper base of each trapezoid is half the length of the lower base. We will use the method of integration to find the volume of this solid.

Understanding the Problem

To begin, let's visualize the problem. We have a solid with a base bounded by the curves $x = y^2$ and $x = 4$ . This means that the base of the solid is a region in the $xy$ -plane that is bounded by these two curves. The cross-sections of the solid are taken perpendicular to the $y$ -axis, which means that they are parallel to the $x$ -axis. Each cross-section is a trapezoid with a height of $h = 2$ , and the upper base of each trapezoid is half the length of the lower base.

Setting Up the Integral

To find the volume of the solid, we need to set up an integral that represents the sum of the volumes of the trapezoidal cross-sections. The volume of each trapezoid can be represented as $\frac{1}{2}h(b_1 + b_2)$ , where $h$ is the height of the trapezoid, and $b_1$ and $b_2$ are the lengths of the lower and upper bases, respectively.

In this case, the height of each trapezoid is given as $h = 2$ , and the upper base is half the length of the lower base. Therefore, the volume of each trapezoid can be represented as $\frac{1}{2}(2)(b_1 + \frac{1}{2}b_1)$ .

Finding the Lengths of the Bases

To find the lengths of the bases of the trapezoids, we need to find the $y$ -coordinates of the points where the curves $x = y^2$ and $x = 4$ intersect the $y$ -axis. The $y$ -axis is the line $x = 0$ , so we need to find the $y$ -coordinates of the points where the curves intersect this line.

The curve $x = y^2$ intersects the $y$ -axis at the point $(0, 0)$ , and the curve $x = 4$ intersects the $y$ -axis at the point $(0, 2)$ . Therefore, the length of the lower base of each trapezoid is $b_1 = 2y$ , and the length of the upper base is $b_2 = \frac{1}{2}b_1 = y$ .

Setting Up the Integral

Now that we have found the lengths of the bases of the trapezoids, we can set up the integral that represents the sum of the volumes of the trapezoidal cross-sections. The integral is given by:

V = \int_{0}^{2} \frac{1}{2}(2)(2y + y) dy

Evaluating the Integral

To evaluate the integral, we need to follow the order of operations. First, we need to simplify the integrand by combining the terms:

V = \int_{0}^{2} (4y + 2y) dy

V = \int_{0}^{2} 6y dy

Next, we need to integrate the function $6y$ with respect to $y$ . The integral of $6y$ is given by:

\int 6y dy = 3y^2 + C

where $C$ is the constant of integration.

Applying the Fundamental Theorem of Calculus

Now that we have found the antiderivative of the function $6y$ , we can apply the Fundamental Theorem of Calculus to evaluate the definite integral. The Fundamental Theorem of Calculus states that:

\int_{a}^{b} f(x) dx = F(b) - F(a)

where $F(x)$ is the antiderivative of $f(x)$ .

In this case, we have:

V = \int_{0}^{2} 6y dy

V = 3(2)^2 - 3(0)^2

V = 12

Conclusion

In this article, we have found the volume of a solid with trapezoidal cross-sections. The solid has a base bounded by the curves $x = y^2$ and $x = 4$ , and the cross-sections are taken perpendicular to the $y$ -axis. The height of each cross-section is given as $h = 2$ , and the upper base of each trapezoid is half the length of the lower base. We have used the method of integration to find the volume of this solid, and we have found that the volume is equal to $12$ .

References

[1] "Calculus" by Michael Spivak
[2] "Calculus: Early Transcendentals" by James Stewart
[3] "Mathematics for Computer Science" by Eric Lehman, F Thomson Leighton, and Albert R Meyer

Glossary

Trapezoid: A quadrilateral with two pairs of parallel sides.
Cross-section: A two-dimensional slice of a three-dimensional object.
Volume: The amount of space inside a three-dimensional object.
Integration: The process of finding the antiderivative of a function.
Fundamental Theorem of Calculus: A theorem that states that the definite integral of a function can be evaluated using the antiderivative of the function.

Q: What is the formula for finding the volume of a solid with trapezoidal cross-sections?

A: The formula for finding the volume of a solid with trapezoidal cross-sections is given by:

V = \int_{a}^{b} \frac{1}{2}h(b_1 + b_2) dy

where $h$ is the height of the trapezoid, $b_1$ and $b_2$ are the lengths of the lower and upper bases, respectively, and $a$ and $b$ are the limits of integration.

Q: How do I find the lengths of the bases of the trapezoids?

A: To find the lengths of the bases of the trapezoids, you need to find the $y$ -coordinates of the points where the curves intersect the $y$ -axis. The $y$ -axis is the line $x = 0$ , so you need to find the $y$ -coordinates of the points where the curves intersect this line.

Q: What is the relationship between the height of the trapezoid and the lengths of the bases?

A: The height of the trapezoid is given as $h = 2$ , and the upper base is half the length of the lower base. Therefore, the length of the lower base is $b_1 = 2y$ , and the length of the upper base is $b_2 = \frac{1}{2}b_1 = y$ .

Q: How do I evaluate the integral to find the volume of the solid?

A: To evaluate the integral, you need to follow the order of operations. First, you need to simplify the integrand by combining the terms. Then, you need to integrate the function with respect to $y$ . Finally, you need to apply the Fundamental Theorem of Calculus to evaluate the definite integral.

Q: What is the Fundamental Theorem of Calculus?

A: The Fundamental Theorem of Calculus is a theorem that states that the definite integral of a function can be evaluated using the antiderivative of the function. It is given by:

\int_{a}^{b} f(x) dx = F(b) - F(a)

where $F(x)$ is the antiderivative of $f(x)$ .

Q: How do I apply the Fundamental Theorem of Calculus to evaluate the definite integral?

A: To apply the Fundamental Theorem of Calculus, you need to find the antiderivative of the function and then evaluate it at the limits of integration. In this case, the antiderivative of the function is given by:

\int 6y dy = 3y^2 + C

where $C$ is the constant of integration. Therefore, the definite integral is given by:

V = \int_{0}^{2} 6y dy

V = 3(2)^2 - 3(0)^2

V = 12

Q: What is the final answer for the volume of the solid?

A: The final answer for the volume of the solid is $12$ .

Q: What are some common applications of finding the volume of a solid with trapezoidal cross-sections?

A: Some common applications of finding the volume of a solid with trapezoidal cross-sections include:

Engineering: Finding the volume of a solid with trapezoidal cross-sections is used in engineering to calculate the volume of materials needed for construction projects.
Physics: Finding the volume of a solid with trapezoidal cross-sections is used in physics to calculate the volume of objects in motion.
Computer Science: Finding the volume of a solid with trapezoidal cross-sections is used in computer science to calculate the volume of 3D objects in computer graphics.

Q: What are some common mistakes to avoid when finding the volume of a solid with trapezoidal cross-sections?

A: Some common mistakes to avoid when finding the volume of a solid with trapezoidal cross-sections include:

Incorrectly identifying the limits of integration: Make sure to identify the correct limits of integration for the definite integral.
Incorrectly evaluating the integral: Make sure to evaluate the integral correctly using the Fundamental Theorem of Calculus.
Incorrectly applying the Fundamental Theorem of Calculus: Make sure to apply the Fundamental Theorem of Calculus correctly to evaluate the definite integral.

Q: What are some common resources for learning about finding the volume of a solid with trapezoidal cross-sections?

A: Some common resources for learning about finding the volume of a solid with trapezoidal cross-sections include:

Textbooks: Textbooks on calculus and mathematics provide a comprehensive introduction to finding the volume of a solid with trapezoidal cross-sections.
Online resources: Online resources such as Khan Academy, Wolfram MathWorld, and Math Open Reference provide interactive tutorials and examples for finding the volume of a solid with trapezoidal cross-sections.
Practice problems: Practice problems and exercises can help you develop your skills in finding the volume of a solid with trapezoidal cross-sections.