Application of Integrals Case Study Questions

Application of Integrals Case Study Questions Class 12

Application of Integrals — Math Case Study Questions (Free Online Test)

Application of Integrals Case Study Questions

The Application of Integrals Case Study Questions are essential for mastering concepts in calculus. They often appear in math case study questions, especially for Class 12 board exams. With these problems, students learn how integrals are used to calculate areas and volumes.

Importance for Class 12 Students

Practicing math case study questions for class 12 helps improve analytical skills. Moreover, it strengthens problem-solving techniques needed for competitive exams. Because integrals link theory to practical applications, they are crucial for exam success.

Topics Covered in the Online Test

Our free online test includes class 12 math case study questions on topics like area between curves and volumes of revolution. Additionally, it offers step-by-step solutions, making preparation smoother and more effective for every learner.

Case Study:3

Ritika is designing a logo for a cultural event, and she wants the logo to represent a leaf-like symmetric shape. She models one side of the leaf using the curve \( y = \sqrt{1 – x^2} \), which is a semi-circle of radius 1 centered at the origin. She wants to calculate the exact area of this semi-circular region bounded above by the curve and below by the $x$-axis. Ritika needs the area to decide on the cost of etching this design using laser cutting. Using integral calculus, she knows that the area under a curve can be computed by a definite integral over a given interval. In this problem, she uses her understanding of symmetry and definite integration to find the desired area.

Concepts and Formulae Used:

  • Area under a curve: \[ \text{Area} = \int_a^b f(x) \, dx \]
  • Symmetry: The function \( y = \sqrt{1 – x^2} \) is even, so: \[ \int_{-1}^{1} f(x) \, dx = 2 \int_0^1 f(x) \, dx \]
  • This curve represents the upper half of a circle of radius 1.

MCQ Questions:

  1. 1. What type of curve is represented by \( y = \sqrt{1 – x^2} \)?
    Answer: (c)
    Solution: \( y = \sqrt{1 – x^2} \) is the upper half of the circle \( x^2 + y^2 = 1 \)
  2. 2. What are the limits of integration for the area under the curve from left to right?
    Answer: (b)
    Solution: The domain of \( \sqrt{1 – x^2} \) is from \( -1 \) to \( 1 \)
  3. 3. Which of the following expressions represents the area under the curve \( y = \sqrt{1 – x^2} \) from \( x = -1 \) to \( x = 1 \)?
    Answer: (b)
    Solution: Area is given directly by \( \int_{-1}^{1} \sqrt{1 – x^2} \, dx \)
  4. 4. What is the value of \( \int_{-1}^{1} \sqrt{1 – x^2} \, dx \)?
    Answer: (b)
    Solution: The area under \( y = \sqrt{1 – x^2} \) from \( -1 \) to \( 1 \) is the area of a semicircle of radius 1: \[ \text{Area} = \dfrac{1}{2} \pi r^2 = \dfrac{\pi}{2} \]
  5. 5. How does symmetry help in evaluating \( \int_{-1}^{1} \sqrt{1 – x^2} \, dx \)?
    Answer: (b)
    Solution: Since the function is even, the area from \( -1 \) to \( 1 \) is: \[ \int_{-1}^{1} f(x) \, dx = 2 \int_{0}^{1} f(x) \, dx \]

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