Definite Integral: Definition, Properties, and Applications

Question 1

Evaluate the definite integral ∫[0,π/2] sin(x) dx using the Fundamental Theorem of Calculus.

Accepted Answer

1

Answer

-1

Answer

0

Answer

2

Question 2

An object is subjected to a force of F(x) = x^2 newtons over a distance of 5 meters. Calculate the work done by the force.

Accepted Answer

25 J

Answer

12.5 J

Answer

37.5 J

Answer

50 J

Question 3

Calculate the arc length of the curve y = x^3/2 from x = 0 to x = 2.

Accepted Answer

20/3

Answer

10/3

Answer

30/3

Answer

40/3

Question 4

Which integral represents the work done by a force of F(x) = sin(x) newtons over a distance of 2π meters?

Accepted Answer

∫[0,2π] sin(x) dx

Answer

∫[0,2π] cos(x) dx

Answer

∫[0,2π] -sin(x) dx

Answer

∫[0,2π] -cos(x) dx

Question 5

The Fundamental Theorem of Calculus, Part 1, states that if f(x) is continuous on an interval [a, b], then the definite integral of f(x) from a to b is equal to:

Accepted Answer

The area under the curve of f(x) from a to b

Answer

The net change in the function f(x) from a to b

Answer

The average value of f(x) on [a, b]

Answer

The slope of the tangent line to the graph of f(x) at x = a

Question 6

If f(x) is an odd function, then the definite integral ∫[-a, a] f(x) dx is equal to:

Accepted Answer

0

Answer

a

Answer

2a

Answer

Cannot be determined from the given information

Question 7

The definite integral ∫[a, b] f(x) g'(x) dx is equal to:

Accepted Answer

f(b)g(b) - f(a)g(a)

Answer

-[f(b)g(b) - f(a)g(a)]

Answer

f(a)g(b) - f(b)g(a)

Answer

-[f(a)g(b) - f(b)g(a)]

Question 8

Which of the following is the formal definition of the definite integral of a function f(x) over an interval [a, b]?

Accepted Answer

The limit of the sum of the areas of the rectangles that approximate the region under the curve y = f(x) between x = a and x = b as the number of rectangles approaches infinity

Answer

The area bounded by the curve y = f(x), the x-axis, and the vertical lines x = a and x = b

Answer

The difference between the values of the function at the endpoints of the interval

Answer

The average value of the function over the interval

Question 9

Evaluate the definite integral using geometric interpretation:
∫[-1,1] |x| dx

Accepted Answer

2

Answer

0

Answer

1

Answer

4

Question 10

Find the indefinite integral: ∫e^x dx

Accepted Answer

e^x + C

Answer

e^-x + C

Answer

x*e^x + C

Answer

e^x/x + C

Question 11

Determine the integral of the function f(x) = e^x.

Accepted Answer

e^x + C

Answer

xe^x + C

Answer

e^x/x + C

Answer

x^e + C

Question 12

Which of the following integrals represents the area under the curve of y = x^2 from x = 0 to x = 2?

Accepted Answer

∫[0, 2] x^2 dx

Answer

∫[2, 0] x dx

Answer

∫[2, 0] x^2 dx

Answer

∫[0, 2] x dx

Question 13

Calculate the area of the region bounded by the curves y = x^2 and y = 4.

Accepted Answer

∫[0, 2] (4 - x^2) dx

Answer

∫[0, 2] (x^2 - 4) dx

Answer

∫[2, 0] (4 - x^2) dx

Question 14

Evaluate the definite integral ∫[-1, 1] |x| dx.

Accepted Answer

2

Answer

0

Answer

1

Answer

-1

Question 15

Which of the following represents the improper integral ∫[0, ∞] e^(-x) dx?

Accepted Answer

lim[x->∞] ∫[0, x] e^(-t) dt

Answer

∫[∞, 0] e^(-x) dx

Answer

lim[x->0] ∫[0, x] e^(-t) dt

Answer

∫[0, ∞] e^(-x) dx

Question 16

Which of the following statements about the definite integral is incorrect?

Accepted Answer

It can be negative.

Answer

It is a generalization of the Riemann sum.

Answer

It represents the area under a curve.

Answer

It can be used to find the volume of a solid.

Question 17

To evaluate the definite integral ∫[0, 2] x^2 dx using the Fundamental Theorem of Calculus, we need to find its antiderivative first. What is the antiderivative of x^2?

Accepted Answer

(x^3)/3 + C

Answer

x + C

Answer

2x + C

Answer

x^2 + C

Question 18

To find the area of the region bounded by the curves y = x^2 and y = 4, we need to set up a definite integral. What are the limits of integration?

Accepted Answer

[0, 2]

Answer

[-2, 2]

Answer

[-4, 0]

Answer

[0, 4]

Question 19

The arc length of the curve y = √x from x = 0 to x = 4 is given by the definite integral:

Accepted Answer

∫[0, 4] √(1 + (1/4x)) dx

Answer

√∫[0, 4] (1 + (1/4x)) dx

Answer

∫[4, 0] √(1 + (1/4x)) dx

Answer

∫[0, 4] √(1 - (1/4x)) dx

Question 20

The surface area of the surface generated by rotating the curve y = x^2 from x = 0 to x = 2 about the y-axis is given by the definite integral:

Accepted Answer

2π∫[0, 2] x√(1 + (4x^2)) dx

Answer

2π∫[2, 0] x√(1 + (4x^2)) dx

Answer

π∫[2, 0] x√(1 + (4x^2)) dx

Answer

π∫[0, 2] x√(1 + (4x^2)) dx

Question 21

The work done by the force f(x) = 2x + 1 over the distance from x = 0 to x = 3 is given by the definite integral:

Accepted Answer

∫[0, 3] (2x + 1) dx

Answer

-∫[0, 3] (2x + 1) dx

Answer

∫[3, 0] (2x + 1) dx

Answer

-∫[3, 0] (2x + 1) dx

Question 22

What does the definite integral $\int_{a}^{b} f(x) dx$ represent geometrically?

Accepted Answer

The net signed area between the curve of $f(x)$, the x-axis, and the lines $x = a$ and $x = b$.

Answer

The instantaneous rate of change of $f(x)$ at $x = a$.

Question 23

If $F'(x) = f(x)$, what is the value of $\int_{0}^{3} f(x) dx$?

Accepted Answer

$F(3) - F(0)$

Answer

$F'(3) - F'(0)$

Answer

$f(3) - f(0)$

Answer

$f'(3) - f'(0)$

Question 24

Which of the following properties is NOT true for definite integrals?

Accepted Answer

$\int_{a}^{b} f(x) dx = - \int_{b}^{a} f(x) dx$

Answer

$\int_{a}^{b} [f(x) + g(x)] dx = \int_{a}^{b} f(x) dx + \int_{a}^{b} g(x) dx$

Answer

$\int_{a}^{b} c f(x) dx = c \int_{a}^{b} f(x) dx$ (where c is a constant)

Answer

$\int_{a}^{a} f(x) dx = 0$

Question 25

The area bounded by the curves $y = x^2$ and $y = 2x$ is given by:

Accepted Answer

$\int_{0}^{2} (2x - x^2) dx$

Answer

$\int_{0}^{1} (2x - x^2) dx$

Answer

$\int_{0}^{1} (x^2 - 2x) dx$

Answer

$\int_{0}^{2} (x^2 - 2x) dx$

Question 26

If $\int_{2}^{5} f(x) dx = 7$ and $\int_{2}^{8} f(x) dx = 12$, then $\int_{5}^{8} f(x) dx$ equals:

Accepted Answer

5

Answer

19

Answer

-19

Answer

-5