Integrals Case Study Questions Class 12 Maths Chapter 7

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Last Updated on July 21, 2025 by XAM CONTENT

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 7 Integrals.

Chapter	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

Table of Contents

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Case Study Questions on Integrals

Case Study:
A water tank has the shape of an inverted right circular cone, with a height of 10 meters and a base radius of 5 meters. Water is poured into the tank at a variable rate such that the rate of increase of the water height (in meters per hour) at time t hours is given by:

\[ \frac{dh}{dt} = 2 + \sin(\pi t) \]

Initially, the tank is empty (\( h = 0 \) at \( t = 0 \)).

Questions:

Find an expression for the height of water \( h \) (in meters) at any time \( t \) in hours by integrating the given rate.
Compute the height of water in the tank after 2 hours, correct to two decimal places.
Describe how the integral you evaluated models the real-world water filling process.

Solutions:

1. Expression for height of water at any time \( t \):

\[ \frac{dh}{dt} = 2 + \sin(\pi t) \] Integrate both sides with respect to \( t \): \[ \int dh = \int [2 + \sin(\pi t)]\,dt \] \[ h(t) = 2t – \frac{1}{\pi}\cos(\pi t) + C \] Given that \( h(0) = 0 \): \[ 0 = 2 \times 0 – \frac{1}{\pi}\cos(0) + C \implies 0 = 0 – \frac{1}{\pi}(1) + C \implies C = \frac{1}{\pi} \] Thus, \[ \boxed{ h(t) = 2t – \frac{1}{\pi}\cos(\pi t) + \frac{1}{\pi} } \]

2. Height after 2 hours (\( t = 2 \)):

\[ h(2) = 2 \times 2 – \frac{1}{\pi} \cos(2\pi) + \frac{1}{\pi} \] Note that \( \cos(2\pi) = 1 \): \[ = 4 – \frac{1}{\pi} \times 1 + \frac{1}{\pi} = 4 \] Final answer: \[ \boxed{h(2) = 4} \] The height of water in the tank after 2 hours is **4.00 meters**.

3. Real-life model explanation:
Integrating the rate \( \frac{dh}{dt} \) gives the total height of water added over time, considering both the constant and variable components of inflow. This models the actual filling process by summing up the changing rates at every moment, as would occur when filling a tank with a periodically varying flow rate.

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

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Topics from which case study questions may be asked

Integration as Inverse Process of Differentiation
Methods of Integration:
- Substitution
- Partial Fractions
- Integration by Parts
Integration of Algebraic, Trigonometric, Exponential and Logarithmic Functions
Definite Integrals – Properties and Evaluation
Fundamental Theorem of Calculus

Integration by parts is a critical technique for integrating products of functions, especially in algebraic and exponential forms.

Frequently Asked Questions (FAQs) on 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 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.