StudyTunnel Logo
StudyTunnel
by BhamuIntelli Systems LLP

Home

School

CBSE

Class 12 Science (PCM)

Mathematics Part 2

Application of Integrals

Application of Integrals

Action PlanNotesQuestionsMock-Test
Study Action Plan (Sorted by Difficulty)
Area Bounded by a Parabola and Coordinate Axix
Accuracy: 0%
Total Attempted0
Total Correct0
Total Incorrect0
Difficulty Level:
High
Area Under Trigonometric Curves
Accuracy: 0%
Total Attempted0
Total Correct0
Total Incorrect0
Difficulty Level:
High
Elementary Area Strip Concept
Accuracy: 0%
Total Attempted0
Total Correct0
Total Incorrect0
Difficulty Level:
High
Integration Techniques for Complex Curves
Accuracy: 0%
Total Attempted0
Total Correct0
Total Incorrect0
Difficulty Level:
High
Absolute Area vs Algebraic Integral
Accuracy: 0%
Total Attempted2
Total Correct0
Total Incorrect2
Difficulty Level:
High
Area Split Across the x-axis
Accuracy: 0%
Total Attempted2
Total Correct0
Total Incorrect2
Difficulty Level:
High
Area Under x = g(y) Using Horizontal Strips
Accuracy: 0%
Total Attempted2
Total Correct0
Total Incorrect2
Difficulty Level:
High
Area Under y = f(x) Using Vertical Strips
Accuracy: 0%
Total Attempted3
Total Correct0
Total Incorrect3
Difficulty Level:
High
Choosing Limits of Integration from Boundaries
Accuracy: 0%
Total Attempted3
Total Correct0
Total Incorrect3
Difficulty Level:
High
Standard Curve Forms for Area Problems
Accuracy: 0%
Total Attempted6
Total Correct0
Total Incorrect6
Difficulty Level:
High
Horizontal and Vertical Strips in Integration
Accuracy: 17%
Total Attempted6
Total Correct1
Total Incorrect5
Difficulty Level:
High
Definite Integral as Area Under Curve
Accuracy: 33%
Total Attempted6
Total Correct2
Total Incorrect4
Difficulty Level:
High
Area Enclosed by a Circle
Accuracy: 38%
Total Attempted8
Total Correct3
Total Incorrect5
Difficulty Level:
High
Area Enclosed by an Ellipse
Accuracy: 40%
Total Attempted5
Total Correct2
Total Incorrect3
Difficulty Level:
High
Area Bounded by Curves and Lines
Accuracy: 50%
Total Attempted2
Total Correct1
Total Incorrect1
Difficulty Level:
Medium
Historical Development of Integral Calculus
Accuracy: 50%
Total Attempted2
Total Correct1
Total Incorrect1
Difficulty Level:
Medium
Positive and Negative Area Treatment
Accuracy: 50%
Total Attempted4
Total Correct2
Total Incorrect2
Difficulty Level:
Medium
Sketching Region Before Integration
Accuracy: 50%
Total Attempted2
Total Correct1
Total Incorrect1
Difficulty Level:
Medium
Symmetry in Area Calculations
Accuracy: 50%
Total Attempted8
Total Correct4
Total Incorrect4
Difficulty Level:
Medium

Learning Objectives

Top

Learning Objectives

  • Understand the application of integrals in calculating areas under curves.
  • Identify the fundamental theorem of calculus and its role in evaluating definite integrals.
  • Calculate the area bounded by curves and lines using integrals.
  • Analyze the relationship between curves and the x-axis in the context of area calculation.
  • Apply integration techniques to find areas of complex shapes, including circles, ellipses, and parabolas.
  • Recognize the historical development of integral calculus and its foundational concepts.

Practice, Analyze & Improve

Dont just read—test your knowledge! Unlock the Student Workspace to take chapter tests and get instant performance insights.

Deep Analytics

Peer Comparison

Track Progress

Claim Free Account100% Free • No credit card required

Revision Notes & Summary

Top

Chapter 8: Application of Integrals

8.1 Introduction

  • Mathematics helps conceive nature in harmonious form.
  • Geometry provides formulae for areas of simple figures (triangles, rectangles, trapezias, circles).
  • These formulae are inadequate for areas enclosed by curves, requiring Integral Calculus.
  • This chapter focuses on:
    • Area under simple curves
    • Area between lines and arcs of circles, parabolas, and ellipses.

8.2 Area under Simple Curves

  • The area under the curve y = f(x) between x = a and x = b is calculated using definite integrals.
  • The area of an elementary strip is given by:
    • dA = y dx
  • The total area A is expressed as:
    • A = ∫[a to b] f(x) dx

Examples

  1. Area under y = x² from x = 1 to x = 2
  2. Area under y = x⁴ from x = 1 to x = 5
  3. Area bounded by y = sin x from x = 0 to x = 2π

Key Formulas

  • Area under the curve y = f(x):
    Area = ∫[a to b] f(x) dx
  • Area between curves x = g(y) and y-axis:
    Area = ∫[c to d] g(y) dy

Historical Note

  • Integral Calculus origins trace back to ancient Greece's method of exhaustion.
  • Key contributors include:
    • Eudoxus and Archimedes (early methods)
    • Newton (theory of fluxions)
    • Leibnitz (definite integral)
    • Cauchy (concept of limits)

Important Diagrams

Fig 8.1: Area under the curve

  • Shows vertical strips under the curve y = f(x).

Fig 8.5: Area enclosed by a circle

  • Circle equation: x² + y² = a².
  • Area calculated using vertical strips.

Fig 8.10: Area under y = cos x from 0 to 2π

  • Shows three shaded regions representing segments of the integral.
Do you belong in the Top 10%?

Our analytics show that most students struggle with specific concepts in Application of Integrals. Take a free diagnostic test to generate your personal Learning DNA profile.

Practice Test – MCQs, True/False

Top

Multiple Choice Questions

A.

There is no difference; both are always non-negative.

B.

The definite integral gives a negative value, while the geometrical area is non-negative.

C.

The definite integral gives a positive value, while the geometrical area is negative.

D.

Both give negative values.
Correct Answer: B

Solution:

When the curve lies below the x-axis, the definite integral results in a negative value, but the geometrical area is always taken as non-negative.

Chapter Concept:

Absolute Area vs Algebraic Integral

A.

12π12\pi

B.

24π24\pi

C.

36π36\pi

D.

48π48\pi
Correct Answer: B

Solution:

The area of the ellipse is π×a×b=π×4×3=12π\pi \times a \times b = \pi \times 4 \times 3 = 12\pi. Since we consider the entire ellipse, the area is 24π24\pi.

Chapter Concept:

Area Bounded by Curves and Lines

A.

∫abf(x) dx\int_{a}^{b} f(x) \, dx

B.

∣∫abf(x) dx∣|\int_{a}^{b} f(x) \, dx|

C.

∫ab∣f(x)∣ dx\int_{a}^{b} |f(x)| \, dx

D.

∫abf(x) dy\int_{a}^{b} f(x) \, dy
Correct Answer: B

Solution:

When the curve lies below the x-axis, the definite integral ∫abf(x) dx\int_{a}^{b} f(x) \, dx is negative, so the area is the absolute value ∣∫abf(x) dx∣|\int_{a}^{b} f(x) \, dx|.

Chapter Concept:

Definite Integral as Area Under Curve

A.

πab\pi a b

B.

πa2b2\pi a^2 b^2

C.

2πab2 \pi a b

D.

πab2\frac{\pi a b}{2}
Correct Answer: A

Solution:

The area enclosed by an ellipse is given by the formula πab\pi a b, where aa and bb are the semi-major and semi-minor axes, respectively.

Chapter Concept:

Area Enclosed by an Ellipse

A.

ydxy dx

B.

xdyx dy

C.

ydyy dy

D.

xdxx dx
Correct Answer: A

Solution:

The area of an elementary vertical strip is given by ydxy dx where yy is the height and dxdx is the width.

Chapter Concept:

Elementary Area Strip Concept

A.

The area can be calculated by integrating over one quadrant and multiplying by 4.

B.

The area is calculated by integrating over the entire circle directly.

C.

The area can be calculated by integrating over one half of the circle and multiplying by 2.

D.

The area is calculated by integrating over one eighth of the circle and multiplying by 8.
Correct Answer: A

Solution:

Due to the symmetry of the circle about both the x-axis and y-axis, the area can be calculated by integrating over one quadrant and multiplying by 4.

Chapter Concept:

Symmetry in Area Calculations

A.

The integral bounds are determined by the x-coordinates where the parabola intersects the lines.

B.

The integral bounds are determined by the y-coordinates where the parabola intersects the lines.

C.

The area is independent of the choice of horizontal or vertical strips.

D.

Horizontal strips cannot be used for this parabola.
Correct Answer: B

Solution:

When using horizontal strips, the integral bounds are determined by the y-coordinates where the parabola intersects the horizontal lines, i.e., y=cy = c and y=dy = d.

Chapter Concept:

Area Bounded by a Parabola and Coordinate Axix

A.

99

B.

1818

C.

2727

D.

3636
Correct Answer: A

Solution:

The area is calculated using horizontal strips. The integral is ∫03x dy=∫03y24 dy=[y312]03=9\int_{0}^{3} x \, dy = \int_{0}^{3} \frac{y^2}{4} \, dy = \left[ \frac{y^3}{12} \right]_{0}^{3} = 9.

Chapter Concept:

Area Bounded by a Parabola and Coordinate Axix

A.

πa2\pi a^2

B.

2πa22\pi a^2

C.

12πa2\frac{1}{2} \pi a^2

D.

4πa24\pi a^2
Correct Answer: A

Solution:

The area of the entire circle is given by the formula A=πa2A = \pi a^2. This is derived by considering the entire angle of the circle, 2π2\pi, in the sector area formula A=12a2θA = \frac{1}{2} a^2 \theta, which results in A=πa2A = \pi a^2 when θ=2π\theta = 2\pi.

Chapter Concept:

Area Enclosed by a Circle

A.

The area decreases as aa increases.

B.

The area remains unchanged as aa increases.

C.

The area increases as aa increases.

D.

The area becomes zero as aa increases.
Correct Answer: C

Solution:

The area is calculated as ∫cdy24a dy\int_{c}^{d} \frac{y^2}{4a} \, dy. As aa increases, the width of the parabola increases, leading to a larger area.

Chapter Concept:

Area Bounded by a Parabola and Coordinate Axix

Experience the StudyTunnel Method

We do not just give you mock tests. We guide you through a gamified, AI-driven learning path designed to guarantee mastery.

Step 1: Practicing the Concepts

Application of Integrals

Step 2: Master Boss Exam
Locked • Full Phase Assessment
Step 3: Redemption Arena
Locked • AI Weakness Tracker

True or False

Correct Answer: False

Solution:

The areas of parts below the x-axis are taken as positive by considering their absolute values.

Chapter Concept :

Integration Techniques for Complex Curves

Correct Answer: True

Solution:

The area of an ellipse can be calculated using symmetry and integration. By integrating the first quadrant area and multiplying by 4, the area is 4∫0abaa2−x2 dx=πab4 \int_{0}^{a} \frac{b}{a} \sqrt{a^2 - x^2} \, dx = \pi ab.

Chapter Concept :

Area Enclosed by an Ellipse

Correct Answer: True

Solution:

The area under a curve that has portions above and below the x-axis is calculated by taking the absolute values of the areas above and below the x-axis and summing them. This ensures that the total area is positive, as areas below the x-axis would otherwise contribute negatively.

Chapter Concept :

Area Split Across the x-axis

Correct Answer: True

Solution:

The method of exhaustion, used by Greek mathematicians like Eudoxus and Archimedes, is considered an early form of integration as it involved calculating areas and volumes through limits.

Chapter Concept :

Historical Development of Integral Calculus

Correct Answer: True

Solution:

Before setting up an integral, sketching the curve and boundary lines helps in identifying intersection points and determining whether to use vertical or horizontal strips, as emphasized in the text.

Chapter Concept :

Sketching Region Before Integration

Correct Answer: True

Solution:

The definite integral ∫abf(x) dx\int_{a}^{b} f(x) \, dx represents the net area between the curve y=f(x)y = f(x) and the x-axis from x=ax = a to x=bx = b. If the curve is below the x-axis, the integral evaluates to a negative value, but the area is taken as the absolute value of this integral.

Chapter Concept :

Definite Integral as Area Under Curve

Correct Answer: True

Solution:

Vertical strips are used when integrating with respect to xx, where the differential area element is dA=y dxdA = y \, dx. This approach is suitable when the region is more easily expressed in terms of xx.

Chapter Concept :

Horizontal and Vertical Strips in Integration

Correct Answer: False

Solution:

The areas below the x-axis are taken as positive when calculating the total area under a curve.

Chapter Concept :

Integration Techniques for Complex Curves

Correct Answer: True

Solution:

The method of exhaustion was used by ancient Greek mathematicians like Eudoxus and Archimedes to solve problems related to calculating areas and volumes, which is essentially a form of integration.

Chapter Concept :

Historical Development of Integral Calculus

Correct Answer: True

Solution:

When the curve y=f(x)y = f(x) lies below the x-axis, the definite integral will yield a negative value. However, the area is considered as the absolute value of this integral.

Chapter Concept :

Area Bounded by Curves and Lines

AI Learning Assistant
Top

I can help you understand Application of Integrals better. Ask me anything!

Summarize the main points of Application of Integrals.
What are the most important terms to remember here?
Explain this concept like I'm five.
Give me a quick 3-question practice quiz.
StudyTunnel Methodology

We dont just provide random mock tests. Our AI-driven, phase-wise learning system is designed to guarantee mastery from your first chapter to your final exam.

1
Master the Concepts

Start with chapter notes, formulas, and common exam mistakes. Practice with thousands of MCQs, True/False, and Descriptive questions with instant solutions.

2
Conquer Boss Exams

Once you master individual chapters, unlock Phase-wise Boss Exams. These milestone tests simulate the actual exam environment to test your retention across multiple subjects.

3
The Redemption Arena

The ultimate learning hack. Our AI tracks every mistake you make and dynamically generates custom tests targeting your weak points until you achieve 100% accuracy.