Chapter Exam

Chapter 8: Application of Integrals — Online MCQ Test

Q: What is the formula for finding the area under a curve y = f(x) between x = a and x = b?

Answer: A. Area = ∫[a to b] y dx — The vertical strips formula for area bounded along the x-axis is Area = ∫[a to b] y dx, where y = f(x).

Q: For finding the area bounded by a curve between y = c and y = d, which formula should be used?

Answer: B. Area = ∫[c to d] x dy — The horizontal strips formula uses Area = ∫[c to d] x dy when integrating with respect to y-axis limits.

Q: Find the area under the curve y = x from x = 0 to x = 2.

Answer: B. 2 — Area = ∫[0 to 2] x dx = [x²/2] from 0 to 2 = 4/2 - 0 = 2.

Q: [Board-Style] A parabola y² = 8x encloses an area with the line x = 2. Draw a rough sketch and calculate the total enclosed area. Which of the following correctly represents the setup?

Answer: A. Area = 2∫[0 to 2] 2√(2x) dx — For y² = 8x at x = 2, y = ±4. Using vertical strips: Area = 2∫[0 to 2] y dx = 2∫[0 to 2] 2√(2x) dx. Evaluating: 4∫[0 to 2] √(2x) dx = 4 × 2√2 × [2x^(3/2)/3]₀² = (16√2/3) × 2^(3/2) = 32/3.

MATHS · CLASS 12 SECOND PUC · Karnataka State Board

Practice Chapter 8: Application of Integrals with a free chapter-wise online MCQ test for Karnataka State Board CLASS 12 SECOND PUC MATHS. This chapter covers: Area Under Curves: The primary objective of this chapter is to calculate the precise geometric area bounded by algebraic and trigonometric curves using definite integrals.Vertical.... AI-generated questions from basic to board-exam level, with instant results and explanations.

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Chapter 8: Application of Integrals — Important Questions & Answers (FAQ)

Frequently asked questions from Karnataka State Board CLASS 12 SECOND PUC MATHS — Chapter 8: Application of Integrals, with answers and explanations. These are sample questions; the exam has its own separate question set.

What is the formula for finding the area under a curve y = f(x) between x = a and x = b?

A. Area = ∫[a to b] y dx ✓
B. Area = ∫[a to b] x dy
C. Area = ∫[a to b] y² dx
D. Area = ∫[a to b] (y dx)²

Answer: A. Area = ∫[a to b] y dx
The vertical strips formula for area bounded along the x-axis is Area = ∫[a to b] y dx, where y = f(x).

For finding the area bounded by a curve between y = c and y = d, which formula should be used?

A. Area = ∫[c to d] y dy
B. Area = ∫[c to d] x dy ✓
C. Area = ∫[c to d] x² dy
D. Area = ∫[c to d] (xy) dy

Answer: B. Area = ∫[c to d] x dy
The horizontal strips formula uses Area = ∫[c to d] x dy when integrating with respect to y-axis limits.

Find the area under the curve y = x from x = 0 to x = 2.

A. 1
B. 2 ✓
C. 3
D. 4

Answer: B. 2
Area = ∫[0 to 2] x dx = [x²/2] from 0 to 2 = 4/2 - 0 = 2.

When a curve crosses the x-axis within the interval [a, b], what strategy should be employed?

A. Ignore the crossing and calculate directly
B. Split the integral at the x-intercept and add absolute values ✓
C. Use only the positive portion
D. Reverse the limits of integration

Answer: B. Split the integral at the x-intercept and add absolute values
The Split Region Strategy requires breaking the integral at x-intercepts to properly account for areas above and below the x-axis.

[Board-Style] A parabola y² = 8x encloses an area with the line x = 2. Draw a rough sketch and calculate the total enclosed area. Which of the following correctly represents the setup?

A. Area = 2∫[0 to 2] 2√(2x) dx ✓
B. Area = ∫[0 to 4] √(y²/8) dy
C. Area = ∫[-4 to 4] 2 dy where x = y²/8
D. Area = 2∫[0 to 4] √(2x) dx

Answer: A. Area = 2∫[0 to 2] 2√(2x) dx
For y² = 8x at x = 2, y = ±4. Using vertical strips: Area = 2∫[0 to 2] y dx = 2∫[0 to 2] 2√(2x) dx. Evaluating: 4∫[0 to 2] √(2x) dx = 4 × 2√2 × [2x^(3/2)/3]₀² = (16√2/3) × 2^(3/2) = 32/3.