1. The Bounded 2D Region
Before revolving, we start with a standard 2D area under a curve. Let's look at the function $f(x) = \sqrt{x}$ on the interval $[0, 4]$.
The area bounded by this curve, the $x$-axis, and the vertical line $x = 4$ forms a flat shape ready to be swept into three dimensions.
The area bounded by this curve, the $x$-axis, and the vertical line $x = 4$ forms a flat shape ready to be swept into three dimensions.
Target Region
$$y = \sqrt{x}, \quad x \in [0, 4]$$
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2. Revolving a Single Slice
Imagine taking a tiny, thin vertical rectangle under the curve at some coordinate $x$. Its width is a microscopic slice, $dx$, and its height is the value of the function, $f(x)$.
When we rotate this single rectangle $360^\circ$ around the $x$-axis, it sweeps out a thin 3D cylinder (a **disk**).
When we rotate this single rectangle $360^\circ$ around the $x$-axis, it sweeps out a thin 3D cylinder (a **disk**).
Revolve the Strip
Volume of One Disk ($dV$)
$$dV = \pi \cdot [r(x)]^2 \cdot dx = \pi \cdot [f(x)]^2 \, dx$$
3. Approximating with $N$ Disks
We can approximate the entire solid's volume by filling it with multiple adjacent disks.
Adjust the count of slices below. Notice that as the number of slices $N$ increases, the blocky approximation begins to match the true curved shape of the continuous solid.
Adjust the count of slices below. Notice that as the number of slices $N$ increases, the blocky approximation begins to match the true curved shape of the continuous solid.
Number of Slices ($N$)
Riemann Approximation
$$V \approx \sum_{i=1}^{N} \pi [f(x_i)]^2 \Delta x$$
4. Calculus & Exact Solution
By taking the limit as the number of slices $N \to \infty$, the slice width $dx \to 0$. The sum becomes a definite integral, calculating the exact volume of the solid.
Let's evaluate the exact volume of our shape revolved around the $x$-axis.
Let's evaluate the exact volume of our shape revolved around the $x$-axis.
Solid Cut-Away View
Integration Steps
1 / 6$$V = \int_{a}^{b} \pi [f(x)]^2 \, dx$$