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Calculus I
Table of Contents
Unit 6 Overview: Applications of Integration
6.1 Areas between Curves
6.2 Determining Volumes by Slicing
6.3 Volumes of Revolution: Cylindrical Shells
6.4 Arc Length of a Curve and Surface Area
6.5 Physical Applications
6.6 Moments and Centers of Mass
6.7 Integrals, Exponential Functions, and Logarithms
6.8 Exponential Growth and Decay
6.9 Calculus of the Hyperbolic Functions

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6.4 Arc Length of a Curve and Surface Area

Citation:

Arc length and surface area calculations are key in understanding curved shapes. These concepts use calculus to measure distances along curves and areas of rotated figures.

Integrating derivatives helps find arc lengths of various function types. For surface area, we rotate curves around axes and integrate. These tools are crucial for analyzing complex shapes in math and real-world applications.

Arc Length and Surface Area

Fundamentals of Calculus for Arc Length and Surface Area

  • Calculus provides essential tools for analyzing curves and surfaces
  • Key concepts include:
    • Functions: Mathematical relationships between variables
    • Derivatives: Measure the rate of change of a function
    • Integration: Process of finding the area under a curve or accumulating quantities

Arc length of y = f(x) curves

  • Measures the distance along a curved line between two points on a graph
  • For a curve defined by $y = f(x)$ on the interval $[a, b]$, the arc length $L$ is calculated using the formula:
    • $L = \int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} dx$
  • Steps to calculate arc length:
    1. Find the derivative of $f(x)$, denoted as $\frac{dy}{dx}$
    2. Substitute $\frac{dy}{dx}$ into the arc length formula
    3. Evaluate the definite integral from $a$ to $b$
  • Example: Calculate the arc length of the curve $y = \frac{1}{3}x^{3/2}$ from $x = 0$ to $x = 8$
    • $\frac{dy}{dx} = \frac{1}{2}\sqrt{x}$
    • $L = \int_{0}^{8} \sqrt{1 + \left(\frac{1}{2}\sqrt{x}\right)^2} dx$
    • Solve the integral to determine the arc length (parabola)

Arc length of x = g(y) curves

  • When a curve is defined by $x = g(y)$ on the interval $[c, d]$, the arc length formula is adjusted to:
    • $L = \int_{c}^{d} \sqrt{1 + \left(\frac{dx}{dy}\right)^2} dy$
  • Steps to calculate arc length:
    1. Find the derivative of $g(y)$, denoted as $\frac{dx}{dy}$
    2. Substitute $\frac{dx}{dy}$ into the modified arc length formula
    3. Evaluate the definite integral from $c$ to $d$
  • Example: Calculate the arc length of the curve $x = \frac{1}{4}y^2$ from $y = 0$ to $y = 2$
    • $\frac{dx}{dy} = \frac{1}{2}y$
    • $L = \int_{0}^{2} \sqrt{1 + \left(\frac{1}{2}y\right)^2} dy$
    • Solve the integral to determine the arc length (quadratic function)

Surface area of rotational solids

  • Generated by rotating a curve $y = f(x)$ on the interval $[a, b]$ around an axis, creating a solid of revolution
  • The surface area $S$ of the solid formed by rotating the curve around the x-axis is given by:
    • $S = 2\pi \int_{a}^{b} f(x) \sqrt{1 + \left(\frac{dy}{dx}\right)^2} dx$
  • The surface area $S$ of the solid formed by rotating the curve around the y-axis is given by:
    • $S = 2\pi \int_{a}^{b} x \sqrt{1 + \left(\frac{dy}{dx}\right)^2} dx$
  • Steps to calculate surface area:
    1. Identify the curve $y = f(x)$ and the interval $[a, b]$
    2. Find the derivative of $f(x)$, denoted as $\frac{dy}{dx}$
    3. Substitute $f(x)$ and $\frac{dy}{dx}$ into the appropriate surface area formula based on the axis of rotation
    4. Evaluate the definite integral from $a$ to $b$
  • Example: Calculate the surface area of the solid formed by rotating the curve $y = \sqrt{x}$ from $x = 0$ to $x = 1$ around the x-axis
    • $\frac{dy}{dx} = \frac{1}{2\sqrt{x}}$
    • $S = 2\pi \int_{0}^{1} \sqrt{x} \sqrt{1 + \left(\frac{1}{2\sqrt{x}}\right)^2} dx$
    • Solve the integral to determine the surface area (square root function)

Key Terms to Review (6)

Absolute value function: An absolute value function is a function that contains an algebraic expression within absolute value symbols. The output of the absolute value function is always non-negative.
Arc length: Arc length is the distance along a curve between two points. In calculus, it is calculated using integration techniques.
Frustum: A frustum is the portion of a solid (normally a cone or pyramid) that lies between two parallel planes cutting it. It can be generated by slicing the solid with a plane parallel to its base and removing the top portion.
Smooth: A smooth function is one that has continuous derivatives up to the required order over a given interval. In calculus, this typically means the function is differentiable and its derivative is also continuous.
Surface area: Surface area is the measure of the total area that the surface of a three-dimensional object occupies. In calculus, it is often calculated using integration techniques to account for curves and complex shapes.
Function: A function is a mathematical relationship between two or more variables, where one variable (the dependent variable) depends on the value of the other variable(s) (the independent variable(s)). Functions are central to the study of calculus, as they provide the foundation for understanding concepts like limits, derivatives, and integrals.