Lesson test: The integral

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The integral

1.[3p]
Using the definition and $\sum_{k = 1}^{n} k^{3} = [n (n + 1) / 2]^{2}$ , evaluate $\int_{0}^{1} x^{3} d x$ .

CorrectNot quite: 0.25

2.[2p]
What is the right-endpoint Riemann sum for $x^{2}$ on $[0, 1]$ with four strips?

CorrectNot quite: 0.46875

3.[3p]
Estimate $\int_{1}^{2} \frac{d x}{x}$ with a midpoint sum using two strips, to four decimal places.

CorrectNot quite: 0.6857

4.[3p]
What does the Dirichlet function, equal to 1 at rationals and 0 at irrationals, demonstrate?

That some bounded functions have no Riemann integral at all, so integrability must be proved rather than assumed

Correct

That every function has an area under its graph

The answer is: That some bounded functions have no Riemann integral at all, so integrability must be proved rather than assumed

That continuous functions may fail to be integrable

The answer is: That some bounded functions have no Riemann integral at all, so integrability must be proved rather than assumed

That Riemann sums always converge

The answer is: That some bounded functions have no Riemann integral at all, so integrability must be proved rather than assumed

5.[2p]
Every function continuous on a closed bounded interval is integrable on it.

True

Correct

False

The answer is: True

6.[2p]
Given $\int_{1}^{4} f = 10$ and $\int_{1}^{4} g = - 2$ , find $\int_{1}^{4} (3 f - 2 g)$ .

CorrectNot quite: 34

7.[2p]
What is $\int_{0}^{2 π} \sin x d x$ ?

CorrectNot quite: 0

8.[2p]
A velocity changes sign during a journey. What does $\int v d t$ give?

The displacement, since area below the axis counts negatively

Correct

The distance travelled, which is always positive

The answer is: The displacement, since area below the axis counts negatively

The average speed

The answer is: The displacement, since area below the axis counts negatively

The total time

The answer is: The displacement, since area below the axis counts negatively

9.[3p]
Match each property of the definite integral to its statement.

Linearity
Additivity
Orientation
Average value

$\int_{a}^{c} = \int_{a}^{b} + \int_{b}^{c}$
$\int_{b}^{a} f = - \int_{a}^{b} f$
$\int (f + g) = \int f + \int g$
$\frac{1}{b - a} \int_{a}^{b} f$

Show the answer

Linearity: $\int (f + g) = \int f + \int g$ Additivity: $\int_{a}^{c} = \int_{a}^{b} + \int_{b}^{c}$ Orientation: $\int_{b}^{a} f = - \int_{a}^{b} f$ Average value: $\frac{1}{b - a} \int_{a}^{b} f$

1.[3p] Using the definition and ∑k=1nk3=[n(n+1)/2]2, evaluate ∫01x3dx.

2.[2p] What is the right-endpoint Riemann sum for x2 on [0,1] with four strips?

3.[3p] Estimate ∫12dxx with a midpoint sum using two strips, to four decimal places.

4.[3p] What does the Dirichlet function, equal to 1 at rationals and 0 at irrationals, demonstrate?

5.[2p] Every function continuous on a closed bounded interval is integrable on it.

6.[2p] Given ∫14f=10 and ∫14g=−2, find ∫14(3f−2g).

7.[2p] What is ∫02πsinxdx?

8.[2p] A velocity changes sign during a journey. What does ∫vdt give?

9.[3p] Match each property of the definite integral to its statement.

1.[3p]
Using the definition and $\sum_{k = 1}^{n} k^{3} = [n (n + 1) / 2]^{2}$ , evaluate $\int_{0}^{1} x^{3} d x$ .

2.[2p]
What is the right-endpoint Riemann sum for $x^{2}$ on $[0, 1]$ with four strips?

3.[3p]
Estimate $\int_{1}^{2} \frac{d x}{x}$ with a midpoint sum using two strips, to four decimal places.

4.[3p]
What does the Dirichlet function, equal to 1 at rationals and 0 at irrationals, demonstrate?

5.[2p]
Every function continuous on a closed bounded interval is integrable on it.

6.[2p]
Given $\int_{1}^{4} f = 10$ and $\int_{1}^{4} g = - 2$ , find $\int_{1}^{4} (3 f - 2 g)$ .

7.[2p]
What is $\int_{0}^{2 π} \sin x d x$ ?

8.[2p]
A velocity changes sign during a journey. What does $\int v d t$ give?

9.[3p]
Match each property of the definite integral to its statement.