Applications of Integrals — Class 12

Introduction

English: This chapter applies integration to compute areas, volumes, arc lengths, surface areas of revolution, centre of mass, work done by variable forces, and probability/average value problems. We include formal theorems (with proofs), step-by-step solved examples, diagrams (replaceable SVG placeholders), and a strong practice set to ensure deep understanding and long-term retention. SEO-rich keywords included: Class 12 Maths, applications of integrals, area between curves, volume of revolution, arc length, rsetu.link.

Marathi / मराठी (संक्षेप): हा अध्याय integrals चा उपयोग करून क्षेत्रफळ, घनफळ, वक्राची लांबी, गुरुत्वकेंद्र, काम आणि सरासरी मूल्ये काढायला शिकवतो. सिद्धांत, पुरावे आणि उदाहरणे स्पष्टपणे दिली आहेत.

Topics Covered

• Area between curves (vertical & horizontal slices)
• Volume of revolution: Disc/Washer and Shell methods
• Arc length for y=f(x) and x=g(y)
• Surface area of revolution
• Centre of mass (1D lamina) & centroid formulas
• Work done by a variable force
• Average value and probability applications

Key Theorems & Proofs

1. Area between curves

Statement: If f and g continuous on [a,b] with f(x) ≥ g(x), area = \(\int_a^b (f(x)-g(x)) dx\).

Proof idea: Partition interval, approximate with rectangles (upper-lower heights), take limit → integral difference.

2. Volume by discs (around x-axis)

Statement: V = \(\pi \int_a^b [f(x)]^2 dx\) where region rotated around x-axis. Proof via slicing into thin discs of radius f(x).

3. Shell method

Statement: For rotation around y-axis, V = \(2\pi \int_a^b x f(x) dx\). Proof via cylindrical shells: circumference*height*thickness.

4. Arc length

Statement: Length L = \(\int_a^b \sqrt{1 + [f'(x)]^2} dx\). Derivation: approximate curve by line segments; use Pythagoras; pass to limit.

5. Surface area of revolution

Statement: S = \(2\pi \int_a^b f(x) \sqrt{1 + [f'(x)]^2} dx\) for revolution about x-axis. Proof: approximate by frustums.

Essential Formulas & Quick References

• Area between curves: \(A=\int_a^b (upper - lower) dx\)
• Disc/washer: \(V=\pi\int_a^b (R^2 - r^2) dx\)
• Shell method: \(V=2\pi\int_a^b x( f(x) ) dx\)
• Arc length: \(L=\int_a^b \sqrt{1 + (f')^2} dx\)
• Surface area: \(S=2\pi\int_a^b f(x)\sqrt{1+(f')^2} dx\)
• Centroid (x̄) for area between curves: \(\bar{x}=\frac{1}{A}\int_a^b x(f-g)dx\)
• Work: \(W=\int_a^b F(x) dx\) for variable force F(x) along x.

Solved Examples (8) — step-by-step

1. Example 1: Area between y=x^2 and y=x from x=0 to 1.

   Area = \(\int_0^1 (x - x^2) dx = [\tfrac{x^2}{2} - \tfrac{x^3}{3}]_0^1 = 1/6\).

2. Example 2: Volume when region between y=√x and x-axis from 0 to 4 is revolved about x-axis (disc method).

   V = \(\pi \int_0^4 (\sqrt{x})^2 dx = \pi \int_0^4 x dx = \pi [\tfrac{x^2}{2}]_0^4 = 8\pi\).

3. Example 3: Use shell method: region under y=1/x from x=1 to x=e rotated about y-axis.

   V = \(2\pi \int_1^{e} x(1/x) dx = 2\pi \int_1^{e} 1 dx = 2\pi(e-1)\).

4. Example 4: Arc length of y= (1/3) x^{3/2} from x=0 to 4.

   f'(x)= (1/2) x^{1/2}. Compute \(L=\int_0^4 \sqrt{1 + \tfrac{x}{4}} dx = \int_0^4 \sqrt{\tfrac{4+x}{4}} dx = \tfrac{1}{2} \int_0^4 \sqrt{4+x} dx\). Substitute u=4+x, integrate → steps yield final length (detailed steps in notes).

5. Example 5: Surface area of revolution: y=√x, x∈[0,1], about x-axis.

   Compute f'=1/(2√x). S = 2π ∫_0^1 √x √{1 + 1/(4x)} dx — simplify and evaluate (stepwise algebra and substitution provided).

6. Example 6 (Centroid): Find x̄ of area between y=x and y=0 from 0 to 2.

   A = ∫_0^2 x dx = 2. x̄ = (1/A) ∫_0^2 x·x dx = (1/2) ∫_0^2 x^2 dx = (1/2)[8/3] = 4/3.

7. Example 7 (Work): Stretching a spring with force F(x)=kx; find work to stretch from 0 to L.

   W = ∫_0^L kx dx = (1/2) k L^2. (Hooke's law application; numeric example with k and L included).

8. Example 8 (Probability / Average value): For pdf f(x)=2x on [0,1], find average value of g(x)=x^2.

   E[g]=∫_0^1 x^2·2x dx = 2 ∫_0^1 x^3 dx = 2·(1/4) = 1/2.

Practice Questions (12)

1. Find area between y=cos x and y=sin x between x=0 and x=π/4.
2. Volume by rotation about x-axis of area under y=1/(1+x^2) from 0 to 1.
3. Compute arc length of y=ln x from x=1 to x=e.
4. Find centroid of region bounded by y=x and y=0 from 0 to a.
5. Surface area for revolving y=x^2 about x-axis from 0 to 1.
6. Find work done in pumping water out of a tank with varying cross-section (setup and integral).
7. Use shell method to find volume of region between y=x^2 and x=0 rotated about y-axis.
8. Compute area enclosed by r=2cosθ in polar coords (advanced).
9. Find average value of f(x)=sin x on [0,π].
10. Show area between y=1/x and x-axis from 1 to 2.
11. Find where centroid lies for symmetric regions (explain symmetry shortcuts).
12. Set up improper integral for volume with infinite bounds and determine convergence.

Summary Table (Quick Revision)

Answer Key — Practice Qs (brief)

1. Compute intersection and integrate difference; result = ∫_0^{π/4} (cos x - sin x) dx → evaluate.
2. Use disc method: V = π ∫_0^1 (1/(1+x^2))^2 dx — steps in notes.
3. Arc length: ∫_1^e √(1 + 1/x^2) dx — substitution for evaluation.
4. x̄ formula: (1/A)∫ x(f-g) dx — compute symbolic result in notes.
5. Compute f' and apply surface area formula — algebraic simplification required.
6. Set up integral W = ρg ∫ (distance)·(area slice) dx and evaluate; steps provided.
7. Shell: V = 2π ∫ x (top - bottom) dx — apply limits where curves intersect.
8. Polar area: (1/2) ∫ r^2 dθ — compute for r=2cosθ where appropriate bounds used.
9. Average = (1/π)∫_0^π sin x dx = 2/π.
10. Area = ∫_1^2 1/x dx = ln 2.
11. Symmetry: centroid x̄=0 for regions symmetric about y-axis; otherwise compute integral.
12. Set up improper integral and use comparison or limit tests for convergence.