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Magazine Chapter 10: Problem 56
Find the volume of the solid obtained by revolving the ellipse \(b^{2} x^{2}+a^{2} y^{2}=a^{2} b^{2}\) about the \(y\)-axis.
Short Answer
Expert verified
The volume is \(\frac{4\pi a^{2} b}{3}\).
Step by step solution
01
Rewrite the Ellipse Equation
The given ellipse equation is \(b^{2}x^{2} + a^{2}y^{2} = a^{2}b^{2}\). We need to find \(x^{2}\) in terms of \(y\) to use as a function for integration. Rearrange the equation to solve for \(x^{2}\), which gives us \(x^{2} = \frac{a^{2}b^{2} - a^{2}y^{2}}{b^{2}}\). This can be simplified to \(x^{2} = \frac{a^{2}(b^{2} - y^{2})}{b^{2}}\).
02
Set up the Integral for Volume
We use the disk method to find the volume of the solid. This method involves integrating the area of circular disks from \(y = -b\) to \(y = b\), the bounds on the y-axis for the ellipse. The radius of each disk is \(|x|\) because the axis of rotation is the \(y\)-axis. So, the area \(A\) of each disk is \(\pi x^{2}\). Thus, the volume \(V\) of the solid is \(V = \int_{-b}^{b} \pi x^{2} \, dy\).
03
Substitute and Simplify
Substitute \(x^{2} = \frac{a^{2}(b^{2} - y^{2})}{b^{2}}\) into the integral: \[ V = \int_{-b}^{b} \pi \left(\frac{a^{2}(b^{2} - y^{2})}{b^{2}}\right) \, dy \]. Factor out the constants \(\frac{\pi a^{2}}{b^{2}}\) to get \[ V = \frac{\pi a^{2}}{b^{2}} \int_{-b}^{b} (b^{2} - y^{2}) \, dy \].
04
Compute the Integral
Evaluate the definite integral: \(\int_{-b}^{b} (b^{2} - y^{2}) \, dy\). Split this into two separate integrals: \(\int_{-b}^{b} b^{2} \, dy - \int_{-b}^{b} y^{2} \, dy\). The first integral is \(b^{2}[y]_{-b}^{b} = 2b^{3}\). The second integral is \([\frac{y^{3}}{3}]_{-b}^{b} = \frac{2b^{3}}{3}\).
05
Find the Final Volume
Combine the evaluated integrals: \(2b^{3} - \frac{2b^{3}}{3} = \frac{4b^{3}}{3}\). Substitute back to get the volume: \[ V = \frac{\pi a^{2}}{b^{2}} \cdot \frac{4b^{3}}{3} = \frac{4\pi a^{2} b}{3} \].
06
Confirm the Result
The formula for the volume around the y-axis due to symmetry and similar forms for ellipsoids can be consistent with this result. Hence, the complete solution for the volume obtained is verified.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Disk Method
The Disk Method is a calculus technique used to find the volume of a solid of revolution. When a region is rotated around an axis, it forms a three-dimensional solid. In this scenario, the Disk Method simplifies calculating the volume by considering the object as composed of numerous thin disks. Each disk is perpendicular to the axis of rotation.
To apply the Disk Method, we need to identify the function that defines the radius of these disks. In our problem, it's necessary to find the function representing half of the ellipse, which essentially translates to deriving the formula for the axis of the ellipse in terms of the other variable.
Once this function is established, the area of a disk which is \( \pi x^{2} \) (since area of a circle is \( \pi r^{2} \)) is integrated along the axis of rotation's bounds. Hence, the integral of the disk's area across its width gives the solid's volume.
To apply the Disk Method, we need to identify the function that defines the radius of these disks. In our problem, it's necessary to find the function representing half of the ellipse, which essentially translates to deriving the formula for the axis of the ellipse in terms of the other variable.
Once this function is established, the area of a disk which is \( \pi x^{2} \) (since area of a circle is \( \pi r^{2} \)) is integrated along the axis of rotation's bounds. Hence, the integral of the disk's area across its width gives the solid's volume.
Ellipse
An ellipse represents a curve on a plane surrounding two focal points. For any point on the ellipse, the sum of the distances to the two foci remains constant. In standard form, an ellipse is described using the equation: \( b^{2}x^{2} + a^{2}y^{2} = a^{2}b^{2}\).
The constants \( a \) and \( b \) denote the semi-major and semi-minor axes respectively. In our problem, we aim to revolve this ellipse around the y-axis and hence, it becomes crucial to express \( x^{2} \) in terms of \( y \).
This requires rearranging the ellipse鈥檚 equation, resulting in a formula that simplifies the integration process. The information derived allows us to apply methods like the Disk Method effectively, forming the foundation of the volume calculation.
The constants \( a \) and \( b \) denote the semi-major and semi-minor axes respectively. In our problem, we aim to revolve this ellipse around the y-axis and hence, it becomes crucial to express \( x^{2} \) in terms of \( y \).
This requires rearranging the ellipse鈥檚 equation, resulting in a formula that simplifies the integration process. The information derived allows us to apply methods like the Disk Method effectively, forming the foundation of the volume calculation.
Definite Integral
A Definite Integral helps in determining quantities such as areas, volumes, and other cumulative values. In the context of rotational solids, it quantifies the final volume by integrating the area of a function over an interval.
The process involves integrating from one boundary of the shape to the other, ensuring that every part of the solid is accounted for.
In this task, the Definite Integral is calculated over the interval from \( -b \) to \( b \), which covers the entire span of the ellipse along the y-axis. By solving this integral, we manage to accumulate the volume occupied by the entire solid. The contribution from every cross-sectional disk along the path via integration culminates in the final solid's volume.
The process involves integrating from one boundary of the shape to the other, ensuring that every part of the solid is accounted for.
In this task, the Definite Integral is calculated over the interval from \( -b \) to \( b \), which covers the entire span of the ellipse along the y-axis. By solving this integral, we manage to accumulate the volume occupied by the entire solid. The contribution from every cross-sectional disk along the path via integration culminates in the final solid's volume.
Symmetry in Solids
Symmetry in solids simplifies the calculation process significantly. A symmetrical object means it is identical on either side of its axis or any plane of symmetry. This uniformity allows mathematicians to evaluate half, or other symmetrical portions of a solid, and then extrapolate this evaluation to the entire structure.
In our exercise, the ellipse exhibit symmetry about the y-axis. Consequently, calculations only require analyzing one side. Post analysis, the final volume can be determined by doubling the calculation or using symmetry properties directly to derive the required solution.
Understanding symmetry not only helps in solving problems more efficiently but also aids in verifying results, ensuring that they're consistent with the geometric structure's expected behavior.
In our exercise, the ellipse exhibit symmetry about the y-axis. Consequently, calculations only require analyzing one side. Post analysis, the final volume can be determined by doubling the calculation or using symmetry properties directly to derive the required solution.
Understanding symmetry not only helps in solving problems more efficiently but also aids in verifying results, ensuring that they're consistent with the geometric structure's expected behavior.
