Volume with Disc Method: Revolving Around the x- or y-Axis

#8.9 Volume with Disc Method: Revolving Around the x- or y-Axis

In AP Calculus, we dive deep into the concept of calculating volume through the disc method, where we revolve shapes around either the x- or y-axis. This method is fundamental in solving problems involving volumes of solids of revolution. Let’s get started!

#🌪️ Volumes of Solids of Revolution

When we talk about finding the volume of a solid of revolution, we're essentially determining how much space the 3D shape occupies. To do this, we take a curve and rotate it around a particular axis, which creates a solid shape. The disc method allows us to calculate the volume of these solids accurately.

#🥏 The Disc Method: X-Axis

The disc method involves slicing the solid into infinitely thin discs perpendicular to the axis of rotation. By summing up the volumes of these discs using definite integrals, we obtain the total volume of the solid.

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A Rotated Solid with a cut-out cross section that is disk-shaped with a width of dx

Courtesy of Cochran Math

To find the volume of a solid rotated around the x-axis, as shown above, we add together the volume of many thin cross-sections! These cross sections of the rotate volume have a width of $dx$ (or approaching 0) and a radius of $f(x))$. As shown above, each cross-section is a super flat cylinder, with a volume of $w \times \pi r^2$, where $w$ is the width and $r$ is the radius. Plugging in our formulas for width and radius, we get:

$$\pi (f(x))^2 dx$$

To add many of these volumes together, we use an integral! Our final integral is:

$$\int_{a}^{b}\pi (f(x))^2dx$$

Where $a$ and $b$ are the stated boundaries or endpoints for the given equation $f(x)$, given as $x = a$ and $x=b$.

#🥏 The Disc Method: Y-Axis

Sometimes, you will be asked to rotate an object around the ...