🎓 CUET-PG MATHEMATICS • INTEGRAL CALCULUS

Table of Contents

∫ INTEGRAL CALCULUS
MASTERCLASS

Concepts • Derivations • Examples • Smart Methods • 50 MCQ Test

From the first idea of integration to double and triple integrals — learn the complete topic step by step with GrandMaster Bikram Sutradhar.

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🌟 CUET-PG Mathematics Integral Calculus Masterclass

Welcome to the Integral Calculus Masterclass by GrandMaster Bikram Sutradhar.

This page is designed for students preparing for CUET-PG Mathematics and for learners who want to build strong university-level calculus fundamentals.

The official NTA Mathematics syllabus includes integration as the inverse process of differentiation, definite integrals and their properties, the Fundamental Theorem of Integral Calculus, double and triple integrals, change of order of integration, and applications involving areas and volumes. :contentReference[oaicite:1]{index=1}

🧠 GrandMaster Learning Rule:Don’t simply memorize an integration formula. Understand → Derive → Apply → Practice → Test

🗺️ Complete Topic-Wise Roadmap

01. Meaning of Integration

Integration as the inverse process of differentiation.

02. Indefinite Integrals

Antiderivatives and constants of integration.

03. Definite Integrals

Limits, evaluation and interpretation.

04. Properties

Important properties of definite integrals.

05. Fundamental Theorem

Connecting differentiation and integration.

06. Double Integrals

Integration over two-dimensional regions.

07. Change of Order

Changing dy dx into dx dy and vice versa.

08. Area Applications

Finding areas using double integration.

09. Surface Applications

Surface-area calculations through integration.

10. Triple Integrals

Integration over three-dimensional regions.

11. Volume Applications

Finding volumes using triple integration.

12. CUET-PG Practice

Conceptual and objective questions.

📘 1. Integration as the Inverse Process of Differentiation

Suppose differentiation takes a function and produces its derivative. Integration reverses this process.

d/dx [F(x)] = f(x)

Therefore:

∫ f(x) dx = F(x) + C

Example 1

Evaluate:

∫ 3x² dx

We know:

d/dx(x³)=3x²

Therefore:

∫3x² dx=x³+C

🎯 Check:Differentiate your answer.If:d/dx[x³+C] = 3x²then your integration is correct.

Power Rule

∫xⁿ dx = xⁿ⁺¹/(n+1)+C

provided n ≠ −1.

Example 2

∫x⁵ dx = x⁶/6 + C

✍️ 2. Important Basic Integration Formulas

Function	Integral
xⁿ	xⁿ⁺¹/(n+1)+C, n≠−1
1/x	ln\|x\|+C
eˣ	eˣ+C
aˣ	aˣ/ln(a)+C
sin x	−cos x+C
cos x	sin x+C
sec²x	tan x+C
cosec²x	−cot x+C
sec x tan x	sec x+C
cosec x cot x	−cosec x+C

⚠️ Important:For the logarithmic formula:

∫ dx/x = ln|x|+C

The absolute value matters when the domain permits both positive and negative x.

🔄 3. Integration by Substitution

Substitution is essentially the reverse of the chain rule.

u=g(x)

Then:

du=g'(x)dx

Example

∫2x(x²+1)⁵ dx

Take:

u=x²+1

Then:

du=2x dx

So:

∫u⁵du

Therefore:

u⁶/6+C

Finally:

(x²+1)⁶/6+C

🧩 4. Integration by Parts

∫u dv = uv − ∫v du

Example

∫x eˣ dx

Take:

u=x

and:

dv=eˣdx

Therefore:

du=dx

v=eˣ

Hence:

∫xeˣdx = xeˣ−∫eˣdx

Therefore:

∫xeˣdx=eˣ(x−1)+C

🧠 Smart selection:For many standard problems, choosing u using the LIATE order can be helpful:Logarithmic → Inverse trigonometric → Algebraic → Trigonometric → Exponential.Use it as a guide, not as an unbreakable rule.

📐 5. Definite Integrals

A definite integral has limits:

∫ₐᵇ f(x)dx

If F'(x)=f(x), then the Fundamental Theorem gives:

∫ₐᵇf(x)dx=F(b)−F(a)

Example

∫₀² x² dx

An antiderivative is:

x³/3

Therefore:

[ x³/3 ]₀²

=8/3−0

=8/3

⭐ 6. Important Properties of Definite Integrals

∫ₐᵃf(x)dx=0

∫ₐᵇf(x)dx=−∫ᵇₐf(x)dx

∫ₐᵇ[f(x)+g(x)]dx = ∫ₐᵇf(x)dx+∫ₐᵇg(x)dx

∫ₐᵇcf(x)dx = c∫ₐᵇf(x)dx

∫ₐᵇf(x)dx = ∫ₐᶜf(x)dx+∫ᶜᵇf(x)dx

Reflection Property

∫₀ᵃf(x)dx = ∫₀ᵃf(a−x)dx

🔥 This property can dramatically simplify certain definite-integral questions when f(x) and f(a−x) combine nicely.

🏛️ 7. Fundamental Theorem of Integral Calculus

This theorem creates the deep connection between differentiation and integration.

F(x)=∫ₐˣf(t)dt

Then, under the usual continuity conditions:

F'(x)=f(x)

Why is this powerful?

Differentiation

Measures instantaneous change.

Integration

Accumulates quantities.

FTC

Connects the two operations.

🧠 GrandMaster Thinking:Whenever you see a definite integral, immediately ask: “Can I find an antiderivative?” If yes, FTC often turns the problem into endpoint evaluation.

========================================================= –>

∫∫ 8. Double Integrals

A double integral extends integration to a two-dimensional region.

∬ᴰ f(x,y)dA

For a rectangular region:

a≤x≤b

and:

c≤y≤d

we may write:

∫ₐᵇ∫𝚌ᵈ f(x,y)dy dx

Example

Evaluate:

∫₀¹∫₀²(x+y)dy dx

First integrate with respect to y:

∫₀²(x+y)dy

=xy+y²/2 |₀²

=2x+2

Now integrate with respect to x:

∫₀¹(2x+2)dx

=[x²+2x]₀¹

🔄 9. Change of Order of Integration

Sometimes a double integral is easier after reversing the order.

For example:

∫₀¹∫ₓ¹ f(x,y)dy dx

The original region is:

0≤x≤1

and:

x≤y≤1

From the region:

0≤y≤1

and:

0≤x≤y

Therefore the reversed integral is:

∫₀¹∫₀ʸ f(x,y)dx dy

⚠️ Golden Rule:Do not reverse the limits mechanically.Draw or mentally identify the region first.Then write the new limits.

📊 10. Area Using Double Integrals

If D is a region in the xy-plane, its area can be represented by:

Area(D)=∬ᴰ1 dA

Example

For the rectangle:

0≤x≤2, 0≤y≤3

Area:

∫₀²∫₀³1 dy dx

First:

∫₀³1dy=3

Then:

∫₀²3dx=6

Therefore:

Area = 6

📐 11. Surface-Area Applications

For a surface represented as:

z=f(x,y)

over a region D, the surface area is:

S= ∬ᴰ √(1+fₓ²+fᵧ²) dA

where:

fₓ=∂f/∂x

and:

fᵧ=∂f/∂y

🎯 The important idea is that a small surface element is not simply dx dy when the surface is tilted. The square-root factor accounts for the geometry of the surface.

∭ 12. Triple Integrals

A triple integral extends integration into three dimensions.

∭ᴱ f(x,y,z)dV

For a rectangular box:

a≤x≤b

c≤y≤d

p≤z≤q

we can write:

∫ₐᵇ∫𝚌ᵈ∫ₚᑫ f(x,y,z) dz dy dx

📦 13. Volume Using Triple Integrals

For a three-dimensional region E, volume is:

Volume(E)=∭ᴱ1 dV

Example

Find the volume of:

0≤x≤2

0≤y≤3

0≤z≤4

Therefore:

V=∫₀²∫₀³∫₀⁴1 dz dy dx

First:

∫₀⁴1dz=4

Then:

∫₀³4dy=12

Finally:

∫₀²12dx=24

Volume = 24 cubic units

🧠 14. GrandMaster Problem-Solving Strategy

Identify the integral.

Is it indefinite, definite, double or triple?

Look for structure.

Check substitution, parts, symmetry or a standard formula.

For multiple integrals, identify the region.

Understand the limits before calculating.

Choose the easiest order.

Especially when the question asks for change of order.

Calculate carefully.

Keep the limits attached to the correct variable.

Check the answer.

Differentiate, estimate or inspect the geometry whenever possible.

🔥 GrandMaster Shortcut Philosophy: “First recognize the pattern; then perform the calculation.”

🔥 CUET-PG INTEGRAL CALCULUS — 50 MCQ TEST

Click an option. You will immediately see whether your answer is correct and receive a short explanation.

Suggested Practice: Try without looking at your notes.

Score: 0/50

1. ∫x² dx equals:

Correct: x³/3 + C.

2. ∫1/x dx is:

Correct: ln|x| + C.

3. ∫cos x dx equals:

Correct: sin x + C.

4. ∫sin x dx equals:

Correct: −cos x + C.

5. ∫sec²x dx equals:

Correct: tan x + C.

6. ∫eˣ dx equals:

Correct: eˣ + C.

7. ∫₀¹ x dx equals:

Correct: [x²/2]₀¹ = 1/2.

8. ∫ₐᵃ f(x) dx equals:

Correct: Both limits are identical, so the integral is zero.

9. Reversing limits gives:

Correct: ∫ₐᵇf dx = −∫ᵇₐf dx.

10. ∫ₐᵇ[f(x)+g(x)]dx equals:

Correct: Integration is linear.

11. ∫2x(x²+1)⁵ dx is:

Correct: Use u=x²+1, du=2x dx.

12. Integration by parts formula is:

Correct: ∫u dv = uv − ∫v du.

13. ∫₀²x²dx equals:

Correct: [x³/3]₀² = 8/3.

14. If F′(x)=f(x), then ∫ₐᵇf(x)dx is:

Correct by the Fundamental Theorem of Calculus.

15. ∫₀ᵃ f(x)dx can be transformed using:

Correct: ∫₀ᵃf(x)dx = ∫₀ᵃf(a−x)dx.

16. A double integral generally integrates over:

Correct: Double integrals naturally describe accumulation over two-dimensional regions.

17. Area of region D can be written as:

Correct: Area(D)=∬ᴰ1 dA.

18. Volume of a three-dimensional region E is:

Correct: Triple integration of 1 gives volume.

19. In ∫∫f(x,y)dy dx, which variable is integrated first?

Correct: The innermost differential dy determines the first integration.

20. The region is essential when changing:

Correct: The geometric region determines the new limits.

21. For z=f(x,y), fₓ denotes:

Correct: fₓ is the partial derivative with respect to x.

22. Surface area for z=f(x,y) involves:

Correct: S=∬√(1+fₓ²+fᵧ²)dA.

23. ∫∫∫1 dV represents:

Correct: Triple integration of 1 gives volume.

24. ∫₀²∫₀³1 dy dx equals:

Correct: It is the area of a 2×3 rectangle, equal to 6.

25. ∫₀²∫₀³∫₀⁴1 dz dy dx equals:

Correct: 2×3×4 = 24.

26. ∫x⁻¹dx equals:

Correct: The power rule has the special exception n=−1.

27. If ∫f(x)dx=F(x)+C, then:

Correct: F is an antiderivative of f.

28. The constant C appears in:

Correct: Indefinite integration represents a family of antiderivatives.

29. Integration by parts is derived from:

Correct: The formula follows from the product rule.

30. If f(x)≥0 on [a,b], then ∫ₐᵇf(x)dx is:

Correct: The integral represents accumulated non-negative area.

31. If f is odd, ∫₋ₐᵃ f(x)dx equals:

Correct: The positive and negative contributions cancel.

32. If f is even, ∫₋ₐᵃf(x)dx equals:

Correct: Symmetry doubles the integral over [0,a].

33. In a double integral, dA commonly represents:

Correct: dA represents a small area element.

34. In a triple integral, dV represents:

Correct: dV represents a small volume element.

35. For a rectangular box, volume is:

Correct: Triple integration reproduces lwh for a rectangular box.

36. Which is a natural use of a double integral?

Correct: ∬1 dA gives area.

37. Which operation is most directly reversed by integration?

Correct: Integration and differentiation are inverse processes under the appropriate conditions.

38. Which method is useful for ∫xeˣdx?

Correct: ∫u dv = uv−∫vdu.

39. To change the order of a double integral, first understand:

Correct: The region determines the new bounds.

40. In ∫ₐᵇ∫𝚌ᵈf(x,y)dy dx, the outer integral is with respect to:

Correct: The outer differential is dx.

41. ∫₀¹∫₀¹1 dy dx equals:

Correct: It is the area of a unit square.

42. ∫₀¹∫₀ˣ 1 dy dx equals:

Correct: Inner integral gives x; ∫₀¹x dx = 1/2.

43. The region 0≤x≤1 and x≤y≤1 can be rewritten as:

Correct: 0≤y≤1 and 0≤x≤y.

44. The volume of the unit cube is:

Correct: 1×1×1=1.

45. If f(x,y)=1, then ∬ᴰf dA represents:

Correct: ∬ᴰ1dA=Area(D).

46. If f(x,y,z)=1, then ∭ᴱf dV represents:

Correct: ∭ᴱ1dV=Volume(E).

47. Which statement about an indefinite integral is correct?

Correct: The +C represents the family of antiderivatives.

48. Which is the best first step when changing integration order?

Correct: Understand the geometric region first.

49. Which theorem connects differentiation and definite integration?

Correct: The Fundamental Theorem of Calculus establishes the central connection between differentiation and integration.

50. The strongest preparation strategy for this topic is:

Correct: Concept + derivation + practice + testing builds durable problem-solving ability.

🏆 Understand Your Test Performance

40–50

Excellent conceptual command. Continue mixed-question practice.

30–39

Good foundation. Strengthen weaker subtopics.

20–29

Review formulas and solved examples before another attempt.

Below 20

Restart from the fundamentals and build the concepts systematically.

⚡ One-Minute Integral Calculus Revision

∫f(x)dx = F(x)+C

∫ₐᵇf(x)dx = F(b)−F(a)

Area = ∬ᴰ1dA

Volume = ∭ᴱ1dV

∫u dv = uv−∫vdu

∫xⁿdx=xⁿ⁺¹/(n+1)+C

🔥 GrandMaster Formula Strategy:Don’t make your notebook a museum of formulas.Make it a map: Formula → Meaning → Derivation → Example → MCQ

❓ Integral Calculus — Student FAQs

What is integration?

Integration is an inverse process of differentiation and also provides a mathematical framework for accumulation and area.

What is the difference between definite and indefinite integration?

An indefinite integral gives a family of antiderivatives and includes +C. A definite integral has specified limits and produces a value under the usual conditions.

Why is the Fundamental Theorem of Calculus important?

It establishes the fundamental relationship between differentiation and definite integration, allowing many definite integrals to be evaluated using antiderivatives.

Why do I need to understand the region in double integrals?

The region determines the limits. This becomes especially important when changing the order of integration.

What does a double integral calculate?

Depending on the integrand, a double integral can represent quantities such as area, mass, accumulated density, and other two-dimensional accumulations.

What does a triple integral calculate?

A triple integral accumulates a quantity throughout a three-dimensional region. With integrand 1, it gives the volume of that region.

How should I prepare Integral Calculus for CUET-PG?

Build the fundamental integration techniques first, then master definite-integral properties and the Fundamental Theorem, followed by double/triple integrals, regions, change of order and applications. Finally, practise objective questions under time pressure.

🚀 Learn With bAstronautWay

Want structured mathematical learning and regular practice? Explore the educational resources of bAstronautWay with GrandMaster Bikram Sutradhar.

📚 Learn

Build concepts from fundamentals.

🧠 Understand

Focus on reasoning and derivations.

✍️ Practise

Work through objective and numerical problems.

🏆 Improve

Use tests to identify and strengthen weak areas.

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CUET-PG Mathematics Integral Calculus Masterclass | Complete Notes, Derivations & 50 MCQ Test | GrandMaster Bikram Sutradhar

∫ INTEGRAL CALCULUS MASTERCLASS

🌟 CUET-PG Mathematics Integral Calculus Masterclass

🗺️ Complete Topic-Wise Roadmap

📘 1. Integration as the Inverse Process of Differentiation

Example 1

Power Rule

Example 2

✍️ 2. Important Basic Integration Formulas

🔄 3. Integration by Substitution

Example

🧩 4. Integration by Parts

Example

📐 5. Definite Integrals

Example

⭐ 6. Important Properties of Definite Integrals

Reflection Property

🏛️ 7. Fundamental Theorem of Integral Calculus

Why is this powerful?

∫∫ 8. Double Integrals

Example

🔄 9. Change of Order of Integration

📊 10. Area Using Double Integrals

Example

📐 11. Surface-Area Applications

∭ 12. Triple Integrals

📦 13. Volume Using Triple Integrals

Example

🧠 14. GrandMaster Problem-Solving Strategy

🔥 CUET-PG INTEGRAL CALCULUS — 50 MCQ TEST

🏆 Understand Your Test Performance

⚡ One-Minute Integral Calculus Revision

❓ Integral Calculus — Student FAQs

🚀 Learn With bAstronautWay

🎓 Master Integral Calculus With GrandMaster Bikram Sutradhar

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∫ INTEGRAL CALCULUS
MASTERCLASS