Applications of the Integrals Case Study Questions Class 12 Maths Chapter 8

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Hello students, we are providing case study questions for class Class 12 Maths. Case study questions are the new question format that is introduced in CBSE board. The resources for case study questions are very less. So, to help students we have created chapterwise case study questions for class Class 12 Maths. In this article, you will find case study questions for cbse class Class 12 Maths chapter 8 Applications of the Integrals.

Chapter	Applications of the Integrals
Type of Questions	Case Study Questions
Nature of Questions	Competency Based Questions
Board	CBSE
Class	Class 12
Subject	Maths
Useful for	Class 12 Studying Students
Answers provided	Yes
Difficulty level	Mentioned
Important Link	Class 12 Maths Chapterwise Case Study

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Table of Contents

Case Study Questions on Applications of the Integrals

Case Study:
A city park has a straight walking path along the river, with the river forming a curve defined by \( y = x^2 \) and the path defined by \( y = x + 2 \). The bridge’s ends are at the points where these two curves intersect. The area between the river and path, bounded by the intersection points, needs to be covered with grass mats.

Questions:

Find the coordinates of the points where the path and the river meet.
Using an integral, calculate the area of the region between the river and the path.
Explain the significance of the area found, in real-life terms.

Solutions:

1. Points of Intersection:

Set \( x^2 = x + 2 \): \[ x^2 – x – 2 = 0 \implies (x – 2)(x + 1) = 0 \implies x = 2 \text{ or } x = -1 \] For \( x = -1: \quad y = (-1)^2 = 1 \). For \( x = 2: \quad y = 2^2 = 4 \). \[ \text{Points of intersection: } (-1,\,1) \text{ and } (2,\,4) \]

2. Area between the curves:

Area between \( y = x + 2 \) (above) and \( y = x^2 \) (below), from \( x = -1 \) to \( x = 2 \): \[ \text{Area} = \int_{-1}^{2}[(x + 2) – x^2]\,dx \] \[ = \int_{-1}^{2} (x + 2 – x^2)\,dx \] \[ = \left[ \frac{x^2}{2} + 2x – \frac{x^3}{3} \right]_{-1}^{2} \] Calculate at \( x = 2 \): \[ \frac{(2)^2}{2} + 2 \times 2 – \frac{(2)^3}{3} = 2 + 4 – \frac{8}{3} = 6 – \frac{8}{3} = \frac{18 – 8}{3} = \frac{10}{3} \] At \( x = -1 \): \[ \frac{(-1)^2}{2} + 2 \times(-1) – \frac{(-1)^3}{3} = \frac{1}{2} – 2 + \frac{1}{3} = (\frac{1}{2} + \frac{1}{3}) – 2 = \frac{5}{6} – 2 = -\frac{7}{6} \] Subtract: \[ \text{Area} = \frac{10}{3} – \left(-\frac{7}{6}\right) = \frac{10}{3} + \frac{7}{6} = \frac{20}{6} + \frac{7}{6} = \frac{27}{6} = \frac{9}{2} \] Thus, \[ \boxed{\text{Area} = \frac{9}{2}} \text{ square units} \]

3. Real-life significance: The calculated area, \( \frac{9}{2} \) square units, represents the actual land between the river and the walking path that must be covered with grass mats. This helps the park management estimate the quantity of materials and plan landscaping effectively.

We hope the given case study questions for Applications of the Integrals Class Class 12 helps you in your learning.

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Topics from which case study questions may be asked

Area under Curves (Standard Curves)
Area Between Two Curves

Finding the area under and between curves helps in visualizing accumulation and has practical use in physics and economics.

Frequently Asked Questions (FAQs) on Applications of the Integrals Case Study Questions

Q1: What is a case study question in mathematics?

A1: A case study question in mathematics is a problem or set of problems based on a real-life scenario or application. It requires students to apply their understanding of mathematical concepts to analyze, interpret, and solve the given situation.

Q2: How should students tackle case study questions in exams?

A2: To tackle case study questions effectively, students should:
Read the problem carefully: Understand the scenario and identify the mathematical concepts involved.
Break down the problem: Divide the case study into smaller parts to manage the information better.
Apply relevant formulas and theorems: Use the appropriate mathematical tools to solve each part of the problem.

Q3: Why are case study questions included in the Class 12 Maths curriculum?

A3: Case study questions are included to bridge the gap between theoretical knowledge and practical application. They help students see the relevance of what they are learning and prepare them for real-life situations where they may need to use these mathematical concepts.

Q4: Are there any online resources or tools available for practicing Applications of the Integrals case study questions?

A12: We provide case study questions for CBSE Class 12 Maths on our website. Students can visit the website and practice sufficient case study questions and prepare for their exams. If you need more case study questions, then you can visit Physics Gurukul website. they are having a large collection of case study questions for all classes.