Free Practice Quiz Question List

09 Definite Integrals and the Fundamental Theorem Online Quiz Questions

Use this free practice quiz with 20 questions to review 09 Definite Integrals and the Fundamental Theorem, test your knowledge, and prepare for your next test or exam.

20 questions

True or false

1 point

True or false: For an integrable function f, reversing the limits gives ∫ from b to a of f(x) dx = −∫ from a to b of f(x) dx.

A
True
B
False

Choose one

1 point

A graph encloses area 7 above the x-axis and area 3 below the x-axis on an interval. What is the value of the definite integral over that interval?

A
4
B
10
C
-4
D
21

Written response

1 point

What is the mathematical term for the sum Σ f(x_i*)Δx used to approximate a definite integral?

Fill in the blank

1 point

In the notation ∫ from a to b of f(x) dx, a and b are the , and f(x) is the .

Choose all

1 point

Select all correct properties of definite integrals.

A
∫ from a to b of [f(x)+g(x)] dx = ∫ from a to b of f(x) dx + ∫ from a to b of g(x) dx
B
∫ from a to b of f(x) dx = ∫ from b to a of f(x) dx
C
∫ from a to b of c f(x) dx = c∫ from a to b of f(x) dx
D
∫ from a to a of f(x) dx = 1

True or false

1 point

True or false: If velocity changes sign during a time interval, integrating velocity over the interval gives net displacement, not necessarily total distance traveled.

A
True
B
False

Choose one

1 point

Let F(x) = ∫ from 1 to x of √(1+t²) dt. What is F'(x)?

A
F'(x) = ∫ from 1 to x of √(1+t²) dt
B
F'(x) = 1/√(1+x²)
C
F'(x) = √(1+x²)
D
F'(x) = 2x

Written response

1 point

A particle has velocity v(t) = t − 2 units per second for 0 ≤ t ≤ 4. What is its net displacement over this interval, in units?

Fill in the blank

1 point

If r(t) is the rate of change of Q(t), then ∫ from a to b of r(t) dt gives the in Q over the interval.

Choose all

1 point

Select all steps that are appropriate for finding the total change represented by a rate r(t) over an interval.

A
Find where the rate is zero in the interval.
B
Determine the sign of the rate on the resulting subintervals.
C
Use the signed integral without regard to sign changes.
D
Integrate the absolute value of the rate, or reverse signs on negative intervals.

Choose one

1 point

Evaluate ∫ from 1 to 3 of (3x² − 4x + 1) dx.

A
6
B
12
C
18
D
24

Choose one

1 point

Let G(x) = ∫ from 2 to x³ of cos(t²) dt. What is G'(x)?

A
cos(x⁶)
B
3x² cos(x⁶)
C
3x cos(x²)
D
cos(3x²)

Open ended

1 point

A tank has net flow rate $r(t)=6-2t$ liters per minute for $0\le t\le5$ . Explain why the net change in the amount of water is 5 liters while the total amount of water moved is 13 liters. Include the relevant integrals and explain the role of the sign change.

Choose one

1 point

A graph encloses an area of 7 square units above the x-axis and 3 square units below the x-axis on an interval. What is the value of the definite integral over that interval?

A
10
B
4
C
-4
D
21

Choose one

1 point

Which statement best explains why the definite integrals $\int_a^b f(x)\,dx$ and $\int_a^b f(t)\,dt$ have the same value?

A
The two integrals always have opposite signs.
B
The first integral is defined, but the second is not.
C
The two integrals are equal because the variable of integration is a placeholder.
D
The two integrals are equal only when f is constant.

Choose one

1 point

If r(t) is the rate of change of a quantity Q(t), what does $\int_a^b r(t)\,dt$ represent?

A
The net change in Q from a to b
B
The average value of Q on [a,b]
C
The total amount of Q present at time b
D
The instantaneous rate of change of Q at time b

Choose one

1 point

Suppose $F(x)=\int_a^x f(t)\,dt$ , where f is continuous. Which expression gives F'(x)?

A
F'(x)=a f(x)
B
F'(x)=f(a)
C
F'(x)=f'(x)
D
F'(x)=f(x)

True or false

1 point

True or false: If a particle's velocity is negative during part of an interval, integrating velocity over the entire interval can give zero displacement even though the particle traveled a positive distance.

A
True
B
False

Written response

1 point

A particle has velocity $v(t)=t-2$ units per second for $0\le t\le4$ . Enter its displacement from time 0 to time 4 as an exact number of units.

Written response

1 point

When differentiating an accumulation function whose upper limit is a function of x, which differentiation rule is used together with the Fundamental Theorem of Calculus? Enter the rule's standard name.