Applications of Definite Integrals: Volume, Average Value & Arc Length

• Compute volumes of revolution using the disk, washer, and shell methods
• Find the average value of a function over an interval
• Calculate arc length of a curve using integration
• Find the surface area of a solid of revolution
• Choose the right method for a given rotation problem

Level 1 — Foundations

When a region is rotated about an axis, it sweeps out a solid. The volume can be found by "slicing" the solid into thin disks (or washers) and summing their volumes.

Disk method (region between f(x) and x-axis, rotated about x-axis): Each disk has radius = f(x) and thickness = dx. Volume of thin disk = π[f(x)]² dx

Average value of f on [a,b]: $f_{\text{avg}} = \frac{1}{b-a}\int_a^b f(x)\,dx$

This is the height of a rectangle on [a,b] with the same area as the region under f.

Level 2 — JEE depth

Disk method (rotation about x-axis): $V = \pi \int_a^b [f(x)]^2\,dx$

Disk method (rotation about y-axis, using x=g(y)): $V = \pi \int_c^d [g(y)]^2\,dy$

Washer method (region between f(x) ≥ g(x) ≥ 0, rotated about x-axis): $V = \pi \int_a^b \left[[f(x)]^2 - [g(x)]^2\right]\,dx$

Shell method (rotation about y-axis, region under f(x) from a to b): $V = 2\pi \int_a^b x \cdot f(x)\,dx$

Shell method is often easier than washer when rotating about y-axis.

Average value of f on [a,b]: $f_{\text{avg}} = \frac{1}{b-a}\int_a^b f(x)\,dx$

Mean Value Theorem for Integrals: There exists c ∈ [a,b] such that f(c) = f_avg (if f is continuous).

Arc length of y=f(x) from x=a to x=b: $L = \int_a^b \sqrt{1 + [f'(x)]^2}\,dx$

Arc length in parametric form x=x(t), y=y(t): $L = \int_{t_1}^{t_2} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2}\,dt$

Surface area of revolution (y=f(x) rotated about x-axis): $S = 2\pi \int_a^b f(x)\sqrt{1 + [f'(x)]^2}\,dx$

Summary table:

Find the volume of the solid formed by rotating y = √x from x=0 to x=4
about the x-axis (disk method), and also about the y-axis (shell method).

--- About x-axis (disk) ---
Step 1: Radius of disk at position x = √x
Step 2: V = π∫₀⁴ (√x)² dx = π∫₀⁴ x dx
Step 3: = π[x²/2]₀⁴ = π(16/2) = 8π cubic units

--- About y-axis (shell method) ---
Step 1: Shell at position x has height f(x) = √x, radius x, thickness dx
Step 2: V = 2π∫₀⁴ x·√x dx = 2π∫₀⁴ x^(3/2) dx
Step 3: = 2π[x^(5/2)/(5/2)]₀⁴
       = 2π · (2/5) · 4^(5/2)
       = 2π · (2/5) · 32
       = 128π/5 cubic units

--- Average value of y = √x on [0,4] ---
f_avg = (1/(4−0))∫₀⁴ √x dx
      = (1/4)[2x^(3/2)/3]₀⁴
      = (1/4)(2·8/3)
      = (1/4)(16/3)
      = 4/3

• Find the volume of the solid formed by rotating y=x² from 0 to 2 about the x-axis
• Find the volume of the sphere of radius r (rotate semicircle y=√(r²−x²) about x-axis)
• Find the average value of f(x)=sin x on [0,π]
• Find the arc length of y=x^(3/2) from x=0 to x=4
• Find the volume generated by rotating the region between y=x and y=x² about the x-axis

• These topics (volume, arc length) are at the JEE extended level; NCERT Class 12 Chapter 8 focuses on area. For volume, refer to JEE-specific resources or Class 12 supplementary material.
• Key diagram: a thin disk (flat cylinder) at position x with radius f(x) and thickness dx
• Key diagram: a thin cylindrical shell at position x with radius x, height f(x), thickness dx

• Volume of cone: rotate y=x from 0 to h about x-axis → V = π∫₀ʰ x² dx = πh³/3 ✓
• Average value of f(x)=x² on [0,3]: → (1/3)∫₀³ x² dx = (1/3)[x³/3]₀³ = 9/3 = 3
• Arc length formula: L = ∫√(1+?) dx → √(1+[f'(x)]²)
• For washer method: V = π∫(R²−r²)dx; which is the outer curve? → The one farther from the axis of rotation
• Stretch: Show that the volume of a sphere of radius r is (4/3)πr³ by integrating with the disk method (rotate semicircle y=√(r²−x²) from −r to r about x-axis).

NCERT Chapter 8 link: Application of Integrals — area is the main focus; volume and arc length are JEE extended topics beyond standard NCERT but core for JEE Main and Advanced
Exam connections: Volume appears in JEE Main 1–2 times per year; average value connects to the Mean Value Theorem (Chapter 5); arc length links to parametric equations and surface area of 3D solids in physics
Study strategy: Master disk method first, then washer (it's just disk with a hole), then shell (a completely different slicing approach). For each, always draw the thin slice before writing the integral.

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