Geometric Applications of Integration in Cartesian Coordinates

1. General Area Formulation as an Integral

Let $f, g: [a, b] \to \mathbb{R}$ be two continuous functions on the closed interval $[a, b]$ such that $f(x) \ge g(x)$ for all $x \in [a, b]$. The area $A$ of the planar region bounded above by $y = f(x)$, below by $y = g(x)$, and laterally by vertical lines $x = a$ and $x = b$ is the Riemann integral of the differential height element:

2. Signed Area vs Total Geometric Area

If the curves intersect multiple times within $[a, b]$ at roots $c_1, c_2, \dots \in (a, b)$, the relative ordering $f(x) \ge g(x)$ flips. The total geometric area is given by the integral of the absolute difference:

3. Horizontal Slicing ($dy$ Integration)

When the boundary curves are more naturally parameterized as functions of $y$, say $x = w_R(y)$ (right boundary) and $x = w_L(y)$ (left boundary) for $y \in [c, d]$ with $w_R(y) \ge w_L(y)$, horizontal slicing avoids piecewise decomposition:

1. Rotational Symmetry & The Differential Disk Element

Let a region bounded by $y = f(x) \ge 0$, the $x$-axis, and lines $x = a, x = b$ be rotated through $2\pi$ radians about the $x$-axis. Slicing perpendicular to the axis of rotation produces circular cross-sectional disks of radius $R(x) = f(x)$ and thickness $dx$.

2. The Washer Method for Annular Cross-Sections

When the planar region is bounded between an outer curve $y = f(x)$ and an inner curve $y = g(x)$ ($0 \le g(x) \le f(x)$), rotation generates an annular washer with outer radius $R(x) = f(x)$ and inner hole radius $r(x) = g(x)$:

3. Rotation About Non-Origin Axes

If the rotation axis is shifted to the horizontal line $y = k$:

Similarly, for rotation about a vertical line $x = h$ using horizontal slices perpendicular to the axis of rotation:

1. When Washer Integration is Pathological

When solving for $x$ in terms of $y$ is algebraically intractable (e.g., $y = 3x^2 - 2x^5$) or requires decomposing the domain into difficult sub-regions, revolving about a vertical axis is vastly superior using cylindrical shells parallel to the axis of revolution.

2. Unrolling the Differential Cylindrical Shell

Consider revolving a thin vertical strip of width $dx$ at distance $x$ from the vertical axis of rotation $x = 0$, having height $h(x) = f(x) - g(x)$. When revolved through $2\pi$, this strip traces a thin hollow cylindrical shell of radius $r = x$, height $h = f(x) - g(x)$, and wall thickness $dx$.

Unrolling this shell into a flat rectangular prism yields the volume element:

3. General Axis of Rotation ($x = h$)

If the axis of rotation is the vertical line $x = h$:

For horizontal axis rotation revolving about $y = k$ with horizontal strips of thickness $dy$:

1. Cavalieri's Principle

If two three-dimensional solids have the same height and equal cross-sectional areas at every cutting plane parallel to their bases, then their total volumes are identical.

2. General Volume by Parallel Slicing

Let a solid extend along the $x$-axis from $x = a$ to $x = b$. If the area of the cross-section perpendicular to the $x$-axis at point $x$ is given by a continuous function $A(x)$, the differential slab volume is $dV = A(x) \, dx$:

3. Common Cross-Sectional Geometry over a Planar Base

Suppose the base of a solid is bounded between $y = f(x)$ and $y = g(x)$, so the cross-sectional base length is $s(x) = f(x) - g(x)$. Standard geometric cross-sections perpendicular to the $x$-axis have areas:

• Squares: $A(x) = [s(x)]^2$
• Equilateral Triangles: $A(x) = \frac{\sqrt{3}}{4} [s(x)]^2$
• Semicircles with diameter $s(x)$: $A(x) = \frac{\pi}{8} [s(x)]^2$
• Isosceles Right Triangles (hypotenuse on base): $A(x) = \frac{1}{4} [s(x)]^2$

1. Differential Arc Length Element $ds$

By the Pythagorean theorem applied to an infinitesimal segment on a smooth curve $y = f(x) \in C^1([a, b])$:

The total Arc Length of the curve from $x = a$ to $x = b$ is:

For a parametric curve $(x(t), y(t))$ for $t \in [t_0, t_1]$, $ds = \sqrt{\dot{x}(t)^2 + \dot{y}(t)^2} \, dt$, so $L = \int_{t_0}^{t_1} \sqrt{\dot{x}^2 + \dot{y}^2} \, dt$.

2. Area of a Surface of Revolution

Revolving the differential arc element $ds$ about an axis sweeps a frustum of a cone of surface area $dS = 2\pi r \, ds$, where $r$ is the perpendicular distance from the curve to the axis of rotation:

• Rotation About the $x$-axis ($r = y = f(x)$): $$\mathbf{S = 2\pi \int_a^b f(x) \sqrt{1 + [f'(x)]^2} \, dx}$$
• Rotation About the $y$-axis ($r = x$): $$\mathbf{S = 2\pi \int_a^b x \sqrt{1 + [f'(x)]^2} \, dx}$$

3. The Centroid Theorems of Pappus

Let a planar curve of length $L$ and centroid distance $\bar{y}$ from an axis in its plane (not crossing the curve) be revolved through $2\pi$:

Let a planar region of area $A$ and centroid distance $\bar{y}$ be revolved through $2\pi$: