Integral Calculus: Master the Basics for Business Applications

Question 1

Compute the area of the region bounded by the curve y = x^2 + 2x and the x-axis between x = 0 and x = 2 using definite integrals.

Accepted Answer

6

Answer

12

Answer

18

Question 2

Revised: The integral of f(x) = x^2 with respect to x is:

Accepted Answer

x^3 + C

Answer

x^3

Answer

2x + C

Answer

2x

Question 3

Revised: Which of the following represents the area under the curve y = f(x) from a to b?

Accepted Answer

∫[a, b] f(x) dx

Answer

∫[b, a] f(x) dx

Answer

∫[0, a] f(x) dx

Answer

∫[0, b] f'(x) dx

Question 4

Revised: The work done by a force F acting over a distance d is:

Accepted Answer

F * d

Answer

1/2 * F * d

Answer

F / d

Answer

1/2 * F / d

Question 5

Revised: Which of the following techniques is NOT used to evaluate improper integrals?

Accepted Answer

Chain Rule

Answer

Comparison Test

Answer

Limit Comparison Test

Answer

Ratio Test

Question 6

Which of the following is NOT an application of integral calculus in business?

Accepted Answer

Predicting future sales based on historical data

Answer

Calculating consumer surplus

Answer

Determining the volume of a cylindrical storage tank

Answer

Estimating the total revenue generated from selling a given quantity of a product

Question 7

Calculate the definite integral that determines the area under the curve of f(x) = 2x + 1 between x = 0 and x = 2.

Accepted Answer

∫[0,2] (2x + 1) dx

Answer

∫[2,0] (2x + 1) dx

Answer

2∫[0,2] (2x + 1) dx

Answer

1/2∫[0,2] (2x + 1) dx

Question 8

Calculate the work done by a force of f(x) = x^3 - 2x^2 + 1 over the distance from x = 0 to x = 2.

Accepted Answer

∫[0,2] (x^3 - 2x^2 + 1) dx

Answer

∫[2,0] (x^3 - 2x^2 + 1) dx

Answer

2∫[0,2] (x^3 - 2x^2 + 1) dx

Answer

1/2∫[0,2] (x^3 - 2x^2 + 1) dx

Question 9

Find the average value of the function f(x) = sin(x) over the interval [0, π/2].

Accepted Answer

1/π

Answer

2/π

Answer

1

Answer

2

Question 10

Calculate the work done by a force of f(x) = e^x over the distance from x = 0 to x = 1.

Accepted Answer

∫[0,1] e^x dx

Answer

∫[1,0] e^x dx

Answer

e∫[0,1] e^x dx

Answer

1/e∫[0,1] e^x dx

Question 11

Find the average value of the function f(x) = x^2 + 1 over the interval [-1, 1].

Accepted Answer

2/3

Answer

1

Answer

1/2

Answer

3/4

Question 12

Prove that the area under the curve y = f(x) from x = a to x = b is equivalent to the definite integral of f(x) from a to b.

Accepted Answer

Use the Fundamental Theorem of Calculus.

Answer

Use the Midpoint Rule.

Answer

Use the Trapezoidal Rule.

Answer

Use Simpson's Rule.

Question 13

The antiderivative of x³ is:

Accepted Answer

x³/3

Answer

x²

Answer

x⁴/4

Answer

x⁵/5

Question 14

Which of the following expressions gives the integral of sin(x) with respect to x?

Accepted Answer

-sin(x) + C

Answer

cos(x)

Answer

-sin(x)

Answer

cos(x) + C

Question 15

The area under the curve of a function f(x) over an interval [a, b] is calculated using the definite integral:

Accepted Answer

∫[a, b] f(x) dx

Answer

∫[b, a] f(x) dx

Answer

∫[a, b] dxf(x)

Answer

∫[b, a] dxf(x)

Question 16

The integral ∫[-1, 2] (x² + 1) dx represents the area under the curve of the function:

Accepted Answer

f(x) = x² + 1

Answer

f(x) = 1 + x²

Answer

f(x) = x² - 1

Answer

f(x) = 1 - x²

Question 17

The integral ∫[0, ∞] e^(-x²) dx:

Accepted Answer

Converges to a finite value

Answer

Diverges to infinity

Answer

Cannot be determined

Answer

Oscillates between positive and negative values

Question 18

To find the volume of a solid of revolution generated by rotating the region bounded by f(x) and g(x) about the x-axis, one can use:

Accepted Answer

Both disk and shell methods

Answer

Disk method only

Answer

Shell method only

Answer

Neither disk nor shell method

Question 19

The work done by a constant force F over a distance s is:

Accepted Answer

Fs

Answer

F/s

Answer

s/F

Answer

1/Fs

Question 20

Which of the following is the formula for the volume of a solid generated by rotating the region bounded by the curves y = x^2 and y = 4 about the x-axis?

Accepted Answer

π∫[0, 2] (4 - x^4) dx

Answer

π∫[0, 2] (x^4 - 4) dx

Answer

2π∫[0, 2] (x^2 - 4) dx

Answer

2π∫[0, 2] (4 - x^2) dx

Question 21

Which of the following is the integral that represents the area of the region bounded by the curves y = sin(x) and y = 0 on the interval [0, π/2]?

Accepted Answer

∫[0, π/2] sin(x) dx

Answer

∫[0, π/2] cos(x) dx

Answer

∫[0, π/2] dx

Answer

-∫[0, π/2] sin(x) dx

Question 22

Find the equation of the line tangent to the curve y = x^3 - 2x^2 + 1 at the point (1, 0).

Accepted Answer

y = x - 1

Answer

y = -x + 1

Answer

y = x - 2

Answer

y = 2x + 1

Question 23

Which of the following is the integral of the function f(x) = e^(2x) with respect to x?

Accepted Answer

(1/2)e^(2x) + C

Answer

e^(2x) + C

Answer

2e^(2x) + C

Answer

(2/e)e^(2x) + C

Question 24

Which of the following is the integral of the function f(x) = cos(x) with respect to x?

Accepted Answer

sin(x) + C

Answer

-cos(x) + C

Answer

cos(x) + C

Answer

-sin(x) + C