Volumes, Surface Areas, Arc Length, and Work: Applications of Definite Integrals

Section 6.1 – Volumes Using Cross-Sections

Volume of Solids with Integrable Cross-Sections

The volume of a solid whose cross-sectional area perpendicular to the x-axis is given by A(x) can be found by integrating A(x) from x = a to x = b. This method is foundational for calculating volumes of irregular solids.

• Key Formula: $V = \int_a^b A(x)\,dx$

• Steps:

  1. Sketch the solid and a typical cross-section.

  2. Find a formula for A(x), the area of a typical cross-section.

  3. Determine the limits of integration.

  4. Integrate A(x) to find the volume.

• Example: Volume of a pyramid with a square base (side 3 m, height 3 m):

  • Cross-sectional area at height x: $A(x) = x^2$

  • Limits: $x = 0$ to $x = 3$

  • Volume: $V = \int_0^3 x^2\,dx = \left.\frac{1}{3}x^3\right|_0^3 = 9$

  • Geometry formula: $V = \frac{1}{3}Bh$

• Example: Volume of a wedge from a cylinder:

  • Cross-sectional area: $A(x) = 9 - x^2$

  • Limits: $x = 0$ to $x = 3$

  • Volume: $V = \int_0^3 (9 - x^2)\,dx$

Solids Formed by Revolution

Volumes of Solids of Revolution

When a function f(x) is rotated about an axis, the resulting solid has circular cross-sections. The area of each cross-section is $A(x) = \pi [f(x)]^2$.

• Key Formula: $V = \int_a^b \pi [f(x)]^2\,dx$

• Example: Volume of a sphere by rotating $x^2 + y^2 = r^2$ about the x-axis:

  • Cross-sectional area: $A(x) = \pi (r^2 - x^2)$

  • Limits: $x = -r$ to $x = r$

  • Volume: $V = \int_{-r}^r \pi (r^2 - x^2)\,dx = \frac{4}{3}\pi r^3$

• Example: Gabriel’s Horn (rotating $y = 1/x$ about the y-axis):

  • Cross-sectional area: $A(y) = \pi/y^2$

  • Limits: $y = -\infty$ to $y = -1$

  • Volume: $V = \int_{-\infty}^{-1} \pi y^{-2}\,dy = \pi$

The Washer Method

Finding Volumes Between Two Surfaces

The washer method is used to find the volume of a solid bounded by two curves when rotated about an axis. The volume is the integral of the difference between the areas of the outer and inner curves.

• Key Formula: $V = \int_a^b \pi [R(x)^2 - r(x)^2]\,dx$

• Example: Rotating $y = x$ and $y = x^2 + 1$ about the x-axis from $x = 0$ to $x = 4$:

  • Outer radius: $R(x) = x^2 + 1$

  • Inner radius: $r(x) = x$

  • Volume: $V = \int_0^4 \pi [(x^2 + 1)^2 - x^2]\,dx$

Section 6.2 – Volumes Using Cylindrical Shells

The Cylindrical Shell Method

This method is useful when the washer method is difficult to apply. It involves integrating the volume of cylindrical shells formed by rotating a region about an axis.

• Key Formula: $V = \int_a^b 2\pi (\text{shell radius}) \times (\text{shell height})\,dx$

• Example: Rotating $y = x$ about the y-axis from $x = 0$ to $x = 4$:

  • Shell radius: $x$

  • Shell height: $x$

  • Volume: $V = \int_0^4 2\pi x x\,dx = 2\pi \int_0^4 x^2\,dx$

• Example: Rotating $y = 3x - x^2$ about $x = -1$ (after shifting):

  • Shell radius: $1 + x$

  • Shell height: $3x - x^2$

  • Volume: $V = \int_0^3 2\pi (1 + x)(3x - x^2)\,dx$

Section 6.3 – Arc Length

Arc Length of a Curve

The arc length of a smooth, differentiable curve $y = f(x)$ from $x = a$ to $x = b$ can be found by integrating the square root of $1 + [f'(x)]^2$.

• Key Formula: $L = \int_a^b \sqrt{1 + [f'(x)]^2}\,dx$

• Example: Find the length of $y = \frac{4}{3}x^{3/2} - 1$ on $[0, 1]$:

  • Compute $f'(x)$ and substitute into the formula.

  • Length: $L = \int_0^1 \sqrt{1 + 8x^{1/2}}\,dx$

Section 6.4 – Areas of Surfaces of Revolution

Surface Area of a Solid of Revolution

The lateral surface area of a solid formed by revolving $y = f(x)$ about the x-axis is given by integrating $2\pi f(x) \sqrt{1 + [f'(x)]^2}$.

• Key Formula: $S = \int_a^b 2\pi f(x) \sqrt{1 + [f'(x)]^2}\,dx$

• Example: Surface area of Gabriel’s Horn ($y = 1/x$ about x-axis, $x \in [1, \infty)$):

  • Surface area: $S = \int_1^{\infty} 2\pi \frac{1}{x} \sqrt{1 + \left(-\frac{1}{x^2}\right)^2}\,dx$

• Example: Surface area of a cone (rotate $y = mx$ about x-axis, $x \in [0, h]$):

  • Surface area: $S = \int_0^h 2\pi m x \sqrt{1 + m^2}\,dx$

  • Result: $S = \pi r \sqrt{h^2 + r^2}$ where $r = mh$

Section 6.5 – Work

Work as an Application of Integration

Work is the energy required to move an object by applying a force over a distance. When the force varies with position, the total work is found by integrating the force function over the displacement.

• Key Formula: $W = \int_a^b F(x)\,dx$

• Example: Lifting fluid from a conical tank:

  • Volume of a thin slice: $\Delta V = \pi r^2 h$

  • Force on each slice: $F(y) = \text{density} \times \Delta V$

  • Distance to lift: $s = 10 - y$ (for a tank of height 10 ft, filled to depth 8 ft)

  • Work for each slice: $\Delta W = 57\pi 4y^2 (10 - y)\Delta y$

  • Total work: $W = \int_0^8 57\pi 4y^2 (10 - y)\,dy$

Summary Table: Methods for Calculating Volumes and Surface Areas

Method	Formula	Typical Use
Cross-Sections	$V = \int_a^b A(x)\,dx$	Solids with known cross-sectional area
Washer Method	$V = \int_a^b \pi [R(x)^2 - r(x)^2]\,dx$	Volume between two surfaces
Cylindrical Shells	$V = \int_a^b 2\pi (\text{radius})(\text{height})\,dx$	Solids of revolution, especially when washer method is difficult
Arc Length	$L = \int_a^b \sqrt{1 + [f'(x)]^2}\,dx$	Length of a curve
Surface Area of Revolution	$S = \int_a^b 2\pi f(x) \sqrt{1 + [f'(x)]^2}\,dx$	Lateral surface area of solids of revolution
Work	$W = \int_a^b F(x)\,dx$	Work done by variable force

Additional info:

• These methods are central to Chapter 6: Applications of Definite Integrals in Calculus.

• Improper integrals (integration to infinity) are introduced in the context of Gabriel’s Horn and will be explored further in later sections.

• Homework problems referenced are typical for practice in these topics.