Class 12 Maths Applications of Integration – NCERT Formulas & Examples

Class 12 Maths Chapter 8: Applications of Integrals

Welcome to FUZY MATH ACADEMY. In this chapter, we study Applications of Integrals like area under curves, volumes of solids of revolution, and average value of a function. This guide contains 25 solved NCERT + IIT-level questions and 15 FAQs with detailed solutions and diagrams where needed.

Basic Rules & Formulas

• ∫ k dx = kx + C
• ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, n ≠ -1
• ∫ e^x dx = e^x + C
• ∫ sin(x) dx = -cos(x) + C
• ∫ cos(x) dx = sin(x) + C
• ∫ u dv = uv - ∫ v du (Integration by Parts)
• ∫ f(g(x)) g'(x) dx = ∫ f(u) du (Substitution)
• Area under y=f(x) from x=a to x=b: ∫[a to b] f(x) dx
• Volume about x-axis: V = π∫[a to b] [f(x)]² dx
• Average value of f(x) on [a,b]: (1/(b-a))∫[a to b] f(x) dx
• Area between curves y=f(x) and y=g(x): ∫[a to b] [f(x)-g(x)] dx
• Volume about y-axis: V = 2π ∫[a to b] x f(x) dx
• Arc length: L = ∫[a to b] √(1 + (dy/dx)²) dx

Examples

Example 1

Find area under y = x² from x=0 to x=2.

Solution: ∫[0 to2] x² dx = [x³/3]_0^2 = 8/3

Example 2

Find the volume of solid generated by rotating y = x² about x-axis from x=0 to x=1.

Solution: V=π∫[0 to1] (x²)² dx = π[x⁵/5]_0^1 = π/5

25 Questions with Solutions

Question 1

∫ (2x+3) dx

x² + 3x + C

Question 2

∫ (x² - 5x + 6) dx

x³/3 - 5x²/2 + 6x + C

Question 3

Area under y = x³ from x=0 to x=2

∫[0 to2] x³ dx = [x⁴/4]_0^2 = 16/4 = 4

Question 4

∫ e^x dx

e^x + C

Question 5

∫ sin(x) dx

-cos(x) + C

Question 6

Volume of solid rotating y = x from x=0 to x=1 about x-axis

V=π∫[0 to1] x² dx = π[x³/3]_0^1 = π/3

Question 7

∫ dx/(1+x²)

tan⁻¹(x) + C

Question 8

∫ (3x² - 2x +1) dx

x³ - x² + x + C

Question 9

Area bounded by y = x² and y = x from x=0 to x=1

∫[0 to1] (x - x²) dx = 1/6

Question 10

Average value of f(x) = x² on [0,2]

(1/2)∫[0 to2] x² dx = 4/3

Question 11

∫ (x+1)² dx

x³/3 + x² + x + C

Question 12

∫ e^(2x) dx

e^(2x)/2 + C

Question 13

Volume of solid rotating y = √x from x=0 to x=4 about x-axis

V=π∫[0 to4] x dx = 8π

Question 14

∫ x e^x dx

By parts: x e^x - e^x + C

Question 15

Area under y = cos(x) from x=0 to x=π/2

∫[0 to π/2] cos(x) dx = 1

Question 16

∫ (2x+1)/(x²+x+1) dx

ln|x²+x+1| + C

Question 17

∫ sec²(x) dx

tan(x) + C

Question 18

∫ (3x²+4x+5) dx

x³ + 2x² + 5x + C

Question 19

Volume of solid rotating y=x² from x=0 to x=1 about y-axis

V=π/2

Question 20

∫ dx/√(1-x²)

sin⁻¹(x) + C

Question 21

∫ (x³ - x) dx

x⁴/4 - x²/2 + C

Question 22

Area between y = x² and y = x³ from x=0 to x=1

∫[0 to1] (x²-x³) dx = 1/6

Question 23

Volume of solid rotating y = x² +1 from x=0 to x=1 about x-axis

V=π∫[0 to1] (x²+1)² dx = 13π/6

Question 24

∫ (sin(x) + cos(x)) dx

-cos(x) + sin(x) + C

Question 25

Average value of f(x) = x³ on [0,2]

(1/2)∫[0 to2] x³ dx = 4

15 FAQs with Solutions

FAQ 1

What is an integral?

Integral represents area under a curve or accumulation of quantities.

FAQ 2

How do I find area under a curve?

Use ∫[a to b] f(x) dx where f(x) is the curve equation.

FAQ 3

What is volume of solid of revolution?

The 3D volume formed by rotating a curve about an axis, calculated using π∫[a to b] [f(x)]² dx.

FAQ 4

How to calculate average value of a function?

(1/(b-a))∫[a to b] f(x) dx

FAQ 5

What is the area between two curves?

∫[a to b] |f(x) - g(x)| dx

FAQ 6

How to find arc length of curve?

L = ∫[a to b] √(1 + (dy/dx)²) dx

FAQ 7

What is integration by parts?

∫ u dv = uv - ∫ v du

FAQ 8

What is substitution method?

Replace inner function u=g(x) to simplify ∫ f(g(x))g'(x) dx → ∫ f(u) du

FAQ 9

How to calculate volume about y-axis?

V = 2π∫[a to b] x f(x) dx

FAQ 10

How to find definite integral?

∫[a to b] f(x) dx = F(b) - F(a) where F(x) is the indefinite integral of f(x)

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