Chapter Exam

Chapter 3: Integral Calculus - II — Online MCQ Test

Q: Which of the following represents the area bounded by the curve $y = f(x)$, the x-axis, and the ordinates $x = a$ and $x = b$ (where $y \geq 0$)?

Answer: A. $ \int_{a}^{b} y \, dx $ — By definition of integration as an area, the area bounded by the curve $y=f(x)$, the x-axis, and the ordinates $x=a$ and $x=b$ is given by $ \int_{a}^{b} y \, dx $.

Q: If the demand function is $p = f(x)$, where $p$ is the price and $x$ is the quantity, what is the formula to calculate the Consumer's Surplus (CS) at equilibrium $(x_0, p_0)$?

Answer: A. $ CS = \int_{0}^{x_0} f(x) \, dx - p_0 x_0 $ — Consumer's Surplus is defined as the total utility (integral of demand function from 0 to $x_0$) minus the actual amount spent ($p_0 x_0$).

Q: Find the area bounded by the curve $y = 3x^2$, the x-axis, and the lines $x = 1$ and $x = 2$.

Answer: A. 7 sq. units — Area = $ \int_{1}^{2} 3x^2 \, dx = [x^3]_1^2 = 2^3 - 1^3 = 8 - 1 = 7 $ sq. units.

Q: Find the area of the region bounded by $y^2 = 4x$ and its latus rectum ($x=1$).

Answer: A. $\frac{8}{3}$ sq. units — Area is symmetric about the x-axis: $2 \int_{0}^{1} \sqrt{4x} \, dx = 4 \int_{0}^{1} x^{1/2} \, dx = 4 [\frac{2}{3} x^{3/2}]_0^1 = \frac{8}{3}$ sq. units.

Q: A firm's marginal cost is given by $MC = 5 + 3e^{-0.03x}$. Find the total cost of producing 100 units if the fixed cost is zero. (Take $e^{-3} \approx 0.05$)

Answer: A. 595 — $C(x) = \int (5+3e^{-0.03x}) \, dx = 5x - 100e^{-0.03x} + k$. Given $C(0) = 0 \implies -100 + k = 0 \implies k = 100$. Thus, $C(100) = 5(100) - 100e^{-3} + 100 = 600 - 100(0.05) = 595$.

BUSINESS MATHS AND STATISTICS · CLASS 12th · Tamil Nadu State Board

Practice Chapter 3: Integral Calculus - II with a free chapter-wise online MCQ test for Tamil Nadu State Board CLASS 12th BUSINESS MATHS AND STATISTICS. This chapter covers: This chapter examines practical applications of definite integration in economics and commerce. Students calculate bounded plane areas consumer surplus producer surplus and total r.... AI-generated questions from basic to board-exam level, with instant results and explanations.

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Chapter 3: Integral Calculus - II — Important Questions & Answers (FAQ)

Frequently asked questions from Tamil Nadu State Board CLASS 12th BUSINESS MATHS AND STATISTICS — Chapter 3: Integral Calculus - II, with answers and explanations. These are sample questions; the exam has its own separate question set.

Which of the following represents the area bounded by the curve $y = f(x)$, the x-axis, and the ordinates $x = a$ and $x = b$ (where $y \geq 0$)?

A. $ \int_{a}^{b} y \, dx $ ✓
B. $ \int_{a}^{b} x \, dy $
C. $ \int_{a}^{b} y^2 \, dx $
D. $ \int_{a}^{b} x^2 \, dy $

Answer: A. $ \int_{a}^{b} y \, dx $
By definition of integration as an area, the area bounded by the curve $y=f(x)$, the x-axis, and the ordinates $x=a$ and $x=b$ is given by $ \int_{a}^{b} y \, dx $.

If the demand function is $p = f(x)$, where $p$ is the price and $x$ is the quantity, what is the formula to calculate the Consumer's Surplus (CS) at equilibrium $(x_0, p_0)$?

A. $ CS = \int_{0}^{x_0} f(x) \, dx - p_0 x_0 $ ✓
B. $ CS = p_0 x_0 - \int_{0}^{x_0} f(x) \, dx $
C. $ CS = \int_{0}^{x_0} f(x) \, dx + p_0 x_0 $
D. $ CS = \int_{0}^{p_0} f(x) \, dx $

Answer: A. $ CS = \int_{0}^{x_0} f(x) \, dx - p_0 x_0 $
Consumer's Surplus is defined as the total utility (integral of demand function from 0 to $x_0$) minus the actual amount spent ($p_0 x_0$).

Find the area bounded by the curve $y = 3x^2$, the x-axis, and the lines $x = 1$ and $x = 2$.

A. 7 sq. units ✓
B. 9 sq. units
C. 3 sq. units
D. 8 sq. units

Answer: A. 7 sq. units
Area = $ \int_{1}^{2} 3x^2 \, dx = [x^3]_1^2 = 2^3 - 1^3 = 8 - 1 = 7 $ sq. units.

Find the area of the region bounded by $y^2 = 4x$ and its latus rectum ($x=1$).

A. $\frac{8}{3}$ sq. units ✓
B. $\frac{4}{3}$ sq. units
C. $\frac{16}{3}$ sq. units
D. $2$ sq. units

Answer: A. $\frac{8}{3}$ sq. units
Area is symmetric about the x-axis: $2 \int_{0}^{1} \sqrt{4x} \, dx = 4 \int_{0}^{1} x^{1/2} \, dx = 4 [\frac{2}{3} x^{3/2}]_0^1 = \frac{8}{3}$ sq. units.

A firm's marginal cost is given by $MC = 5 + 3e^{-0.03x}$. Find the total cost of producing 100 units if the fixed cost is zero. (Take $e^{-3} \approx 0.05$)

A. 595 ✓
B. 605
C. 500
D. 495

Answer: A. 595
$C(x) = \int (5+3e^{-0.03x}) \, dx = 5x - 100e^{-0.03x} + k$. Given $C(0) = 0 \implies -100 + k = 0 \implies k = 100$. Thus, $C(100) = 5(100) - 100e^{-3} + 100 = 600 - 100(0.05) = 595$.