Problem 65
Question
Finding Volumes of a Solid Find the volumes of the solids (see figures) generated if the upper half of the ellipse \(9 x^{2}+25 y^{2}=225\) is revolved about (a) the \(x\) -axis to form a prolate spheroid (shaped like a football), and (b) the \(y\) -axis to form an oblate spheroid (shaped like half of a candy).
Step-by-Step Solution
Verified Answer
The volume of the prolate spheroid will be obtained after integrating the function over the range [-5, 5]. The volume of the oblate spheroid will be obtained after integrating the function over the range [-3, 3]. The volume computation gets updated after each calculation of integral.
1Step 1: Find the y in the ellipse equation
The equation of the ellipse is given as \(9 x^{2}+25 y^{2}=225\). Solving for y to get the equation \(y=\sqrt{(225-9x^{2})/25}\).
2Step 2:Volume of a prolate spheroid
The volume of a prolate spheroid (like a footBall), generated by revolving the given ellipse around x-axis, can be found using the formula: \[V = \pi \int_{{a}}^{{b}} y^{2} dx\] By replacing \(y=\sqrt{(225-9x^{2})/25}\) into the formula, the limits of the integral, which are the x-intercepts can be calculated by solving \(9x^2 = 225\) yielding: \[x=-5, \ 5\] Thus the prolate spheroid volume is: \[V=\pi \int_{-5}^{5}{((225-9x^{2})/25)} dx\] Solving this integral gives the value for the volume.
3Step 3: Volume of an oblate spheroid
Now for the oblate spheroid (like half of a candy), again using the volume formula but revolving around y-axis. So, the formula becomes \[V=2\pi \int_{{c}}^{{d}} x y \ dy\] From ellipse equation \(x=\sqrt{(225-25y^2)/9}\) The limits are the y-intercepts of the ellipse, obtained by solving \(25y^2 = 225\) yielding: \[y=-3, \ 3\] So, the equation becomes \[V=2\pi \int_{-3}^{3}{\sqrt{(225-25y^2)/9}} \times y \ dy.\] Solving this integral gives the volume of the oblate spheroid.
Key Concepts
Prolate SpheroidOblate SpheroidElliptical RevolutionVolume Calculation
Prolate Spheroid
A prolate spheroid is a three-dimensional shape that resembles an elongated sphere, much like a football. It is created by revolving an ellipse around its major axis (the longer axis). This shape is characterized by its long, oval profile.
In mathematics, understanding this shape helps us in various fields such as physics and engineering, especially when dealing with objects that have symmetric, elongated profiles. When an ellipse is rotated around the x-axis, as described in our exercise, it results in a prolate spheroid.
To calculate its volume, we use the calculus-derived formula involving an integral, which helps evaluate the space occupied by this 3D object. Integrating takes into consideration the cross-sectional areas along the path of rotation, providing an exact measure of the volume.
In mathematics, understanding this shape helps us in various fields such as physics and engineering, especially when dealing with objects that have symmetric, elongated profiles. When an ellipse is rotated around the x-axis, as described in our exercise, it results in a prolate spheroid.
To calculate its volume, we use the calculus-derived formula involving an integral, which helps evaluate the space occupied by this 3D object. Integrating takes into consideration the cross-sectional areas along the path of rotation, providing an exact measure of the volume.
Oblate Spheroid
An oblate spheroid is another form of a three-dimensional ellipse, however, it appears flattened at the poles and is more disk-like. This shape is obtained when an ellipse is revolved around its minor axis (the shorter axis), similar to Earth's shape due to its rotation.
Just as with the prolate spheroid, understanding the oblate spheroid is crucial in various scientific domains, such as geophysics, where Earth's slight flattening is considered.
In our exercise, when revolving the given ellipse about the y-axis, the generated shape is an oblate spheroid. To find its volume precisely, we apply a volume integral formula also, but it will differ due to the shape's rotation axis, reflecting a different set of limits and math adjustments that accommodate for the flattening aspect.
Just as with the prolate spheroid, understanding the oblate spheroid is crucial in various scientific domains, such as geophysics, where Earth's slight flattening is considered.
In our exercise, when revolving the given ellipse about the y-axis, the generated shape is an oblate spheroid. To find its volume precisely, we apply a volume integral formula also, but it will differ due to the shape's rotation axis, reflecting a different set of limits and math adjustments that accommodate for the flattening aspect.
Elliptical Revolution
Elliptical revolution involves the rotation of an ellipse around an axis, creating various 3D shapes based on where the axis of rotation is positioned. It’s essentially transforming a 2D elliptical figure into a three-dimensional solid.
Depending on whether the ellipse is revolved around its major or minor axis, different spheroids can be formed, such as prolate or oblate spheroids.
Understanding elliptical revolution helps in visualizing how these solid shapes are generated and allows us to predict their properties, such as the volume or surface area. In the context of our exercise, these revolutions around the x and y axes demonstrate how the shape of a revolving ellipse redefines its dimensional characteristics.
Depending on whether the ellipse is revolved around its major or minor axis, different spheroids can be formed, such as prolate or oblate spheroids.
Understanding elliptical revolution helps in visualizing how these solid shapes are generated and allows us to predict their properties, such as the volume or surface area. In the context of our exercise, these revolutions around the x and y axes demonstrate how the shape of a revolving ellipse redefines its dimensional characteristics.
Volume Calculation
Calculating the volume of solids obtained by elliptical revolution requires integral calculus. This technique offers a precise method to measure the space within these complex shapes.
Using integrals, we consider the infinitesimal slices of the solid and sum these to find the total volume, a process sensitive to the chosen axis of rotation and the resultant shape, whether it be prolate or oblate.
Using integrals, we consider the infinitesimal slices of the solid and sum these to find the total volume, a process sensitive to the chosen axis of rotation and the resultant shape, whether it be prolate or oblate.
- For a prolate spheroid, where the ellipse is rotated about the major axis, the formula \( \pi \int y^2 \, dx \) computes its volume.
- Conversely, an oblate spheroid's volume, calculated by revolving the ellipse around the minor axis, uses the formula \( 2 \pi \int x \, y \, dy \).
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