Unit 8: Applications of Integration

📚Study Guide: Applications of Integration

Unit 8: Applications of Integration

This unit applies definite integrals to compute geometric quantities and solve physical problems. You will find areas between curves, volumes of solids of revolution using disk, washer, and shell methods (AB focuses primarily on disk/washer), and lengths of curves. The concept of accumulating small pieces--whether thin rectangles for area, thin disks for volume, or small line segments for arc length--is central. You will also solve problems involving average value of a function and net change. Understanding how to set up integrals from geometric descriptions is more important than memorizing formulas blindly. The AP exam emphasizes proper integral setup, correct limits, and accurate evaluation.

Key Concepts

• Area Between Curves: Integral of [top function - bottom function] dx or [right function - left function] dy.
• Disk Method: V = pi * integral [R(x)]^2 dx, where R(x) is the radius from the axis of revolution.
• Washer Method: V = pi * integral ([R(x)]^2 - [r(x)]^2) dx, where R is outer radius and r is inner radius.
• Average Value: f_avg = (1/(b-a)) * integral from a to b of f(x) dx.
• Mean Value Theorem for Integrals: There exists c in [a,b] where f(c) equals the average value.
• Arc Length: L = integral from a to b of sqrt(1 + [f'(x)]^2) dx (AB may include setup; BC evaluates).
• Net Change Theorem: Integral of a rate of change gives the net change: integral from a to b of F'(x) dx = F(b) - F(a).

Vocabulary

• Solid of revolution: A solid formed by rotating a region about an axis.
• Disk method: A volume formula treating slices as solid circular disks.
• Washer method: A volume formula treating slices as washers (disks with holes).
• Average value: The constant height of a rectangle with width (b-a) having the same area as the integral.
• Arc length: The distance along a curve between two points.
• Cross-sectional area: The area of a slice perpendicular to an axis, used in volume by slicing.

Formulas

• Area: integral [f(x) - g(x)] dx from a to b (f >= g)
• Disk: V = pi * integral [R(x)]^2 dx
• Washer: V = pi * integral ([R(x)]^2 - [r(x)]^2) dx
• Average value: (1/(b-a)) * integral_a^b f(x) dx
• Arc length: L = integral_a^b sqrt(1 + [f'(x)]^2) dx
• Net change: integral_a^b F'(x) dx = F(b) - F(a)

Common Mistakes

• Using the wrong axis of revolution when setting up R(x) or R(y); always measure radius perpendicular to the axis.
• Subtracting functions in the wrong order for area, yielding a negative result.
• Forgetting to square the radii in disk/washer formulas.
• Confusing average value with average rate of change; they are different concepts.

AP Exam Strategies

• Always draw the region and a representative rectangle/disk before writing the integral; the setup earns most of the credit.
• When revolving around a horizontal line y = k, remember R(x) = |f(x) - k|.
• For area problems with multiple intersections, split the integral at intersection points where top and bottom functions switch.
• Check if the question asks for setup only or evaluation; do not waste time evaluating if only setup is required.

Real-World Applications

• Engineering: Volume calculations design tanks, pipes, and rotational machine parts.
• Medicine: Blood flow models use average velocity across a vessel cross-section.
• Physics: Work done by a variable force is the integral of force over distance.