Free Practice Quiz Question List

6 The Fundamental Theorem of Calculus Online Quiz Questions

Use this free practice quiz with 20 questions to review 6 The Fundamental Theorem of Calculus, test your knowledge, and prepare for your next test or exam.

20 questions

True or false

1 point

True or false: A definite integral represents signed accumulation, so area below the x-axis contributes negatively to the integral.

A
True
B
False

Choose one

1 point

Let $G(x)=\int_1^x\sqrt{1+t^4}\,dt$ . What is $G'(x)$ ?

A
$G'(x)=\sqrt{1+t^4}$
B
$G'(x)=\sqrt{1+x^4}$
C
$G'(x)=\frac{1}{2\sqrt{1+x^4}}$
D
$G'(x)=\sqrt{1+x^4}+C$

Choose all

1 point

Which statements are correct? Select all that apply.

A
If $Q'(t)$ is a rate, then $Q(b)=Q(a)+\int_a^b Q'(t)\,dt$ .
B
Every definite integral equals total geometric area, even when the integrand is negative.
C
If $G(x)=\int_a^x f(t)\,dt$ , then $G'(x)=f(x)$ when the hypotheses of FTC Part 1 hold.
D
For a definite integral, adding the same constant to an antiderivative at both endpoints does not change the result.

Written response

1 point

Find the average value of $f(x)=x^2$ on $[0,3]$ . Enter the numerical value.

Fill in the blank

1 point

If $F'(x)=f(x)$ , complete the Evaluation Theorem: $\int_a^b f(x)\,dx=$ $-$ .

True or false

1 point

True or false: When evaluating a definite integral with an antiderivative, including $+C$ is unnecessary because the constants cancel.

A
True
B
False

Choose one

1 point

A particle has velocity $v(t)=3t^2-4t$ meters per second. What is its displacement from $t=1$ to $t=3$ ?

A
$6\text{ m}$
B
$8\text{ m}$
C
$10\text{ m}$
D
$15\text{ m}$

Written response

1 point

For the area between two curves on an interval where one curve stays above the other, what ordering rule should be used for the integrand?

Fill in the blank

1 point

If $f(x)\ge g(x)$ on $[a,b]$ , complete the area formula: $A=\int_a^b$ $\,dx$ .

Choose one

1 point

Evaluate $\int_1^3(2x^3-4x+1)\,dx$ .

A
$24$
B
$18$
C
$30$
D
$36$

Choose all

1 point

Which are valid applications of integration described in the material? Select all that apply.

A
Position can be recovered from a known velocity.
B
Velocity can be recovered from a known acceleration.
C
Total distance is always equal to the integral of velocity, even when velocity changes sign.
D
The volume of a solid can be found by integrating its cross-sectional area.

Open ended

1 point

Explain how to find the average value of an integrable function on $[a,b]$ , and state what continuity guarantees about where that value occurs.

Written response

1 point

A particle has acceleration $a(t)=2t$ metres per second squared and initial velocity $v(0)=6$ metres per second. Find $v(2)$ . The absolute tolerance is zero.

Choose one

1 point

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

A
$G'(x)=\int_a^x f(t)\,dt$
B
$G'(x)=f(x)$
C
$G'(x)=f(a)$
D
$G'(x)=F(b)-F(a)$

Choose one

1 point

What does $\int_a^b f(x)\,dx$ represent when the graph of $f$ may lie both above and below the $x$ -axis?

A
The total geometric area only
B
The area above the $x$ -axis only
C
The net signed area
D
The perimeter of the region

Choose one

1 point

What is the average value of $f(x)=x^2$ on $[0,3]$ ?

A
3
B
9
C
$\frac{9}{2}$
D
1

Choose one

1 point

If $f(x)\ge g(x)$ on $[a,b]$ , which integral gives the area between their graphs?

A
$\int_a^b[g(x)-f(x)]\,dx$
B
$\int_a^b f(x)g(x)\,dx$
C
$\int_a^b[f(x)+g(x)]\,dx$
D
$\int_a^b[f(x)-g(x)]\,dx$

Choose one

1 point

Which equation correctly expresses the final amount $Q(b)$ when $Q'(t)$ is the rate of change of $Q$ ?

A
$Q(b)=Q(a)-\int_a^b Q'(t)\,dt$
B
$Q(b)=Q(a)+\int_a^b Q'(t)\,dt$
C
$Q(b)=\int_a^b Q(t)\,dt$
D
$Q(b)=Q'(a)+Q'(b)$

True or false

1 point

True or false: If a particle's velocity changes sign on an interval, its total distance traveled can be found by integrating $|v(t)|$ over that interval.

A
True
B
False

Written response

1 point

Evaluate $\int_0^2 3x^2\,dx$ . Enter the numerical value.