Lesson test: What integrals compute

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What integrals compute

1.[3p]
Find the area enclosed between $y = x + 2$ and $y = x^{2}$ .

CorrectNot quite: 4.5

2.[2p]
Find the area enclosed between $y = 2 x$ and $y = x^{2}$ .

CorrectNot quite: 1.333

3.[3p]
Revolving $y = \sqrt{x}$ from $x = 0$ to $x = 4$ about the $x$ -axis gives what volume, to two decimal places?

CorrectNot quite: 25.13

4.[2p]
Integrating discs to get the volume of a cone gives a factor of one third. Where does it come from?

From $\int x^{2} d x = x^{3} / 3$ , the power rule and nothing else

Correct

From the three dimensions of space

The answer is: From $\int x^2 dx = x^3/3$, the power rule and nothing else

From the ratio of the radius to the height

The answer is: From $\int x^2 dx = x^3/3$, the power rule and nothing else

From an approximation that is exact only for a cone

The answer is: From $\int x^2 dx = x^3/3$, the power rule and nothing else

5.[2p]
How much work, in joules, stretches a spring of stiffness 200 N/m by 0.3 m?

CorrectNot quite: 9

6.[3p]
What is the escape velocity from the Earth's surface, in km/s, to one decimal place?

CorrectNot quite: 11.2

7.[2p]
The improper integral $\int_{1}^{\infty} \frac{d x}{x}$ converges.

True

The answer is: False

False

Correct

8.[3p]
A 2 m rod has density $ρ (x) = 1 + x$ kg per metre. Where is its centre of mass, in metres from the light end?

CorrectNot quite: 1.167

9.[2p]
Why is the perimeter of an ellipse a standard example of the limits of integration technique?

The arc length integral it produces has no elementary antiderivative, and the functions invented for it are named after this problem

Correct

The integral diverges

The answer is: The arc length integral it produces has no elementary antiderivative, and the functions invented for it are named after this problem

The ellipse has no arc length

The answer is: The arc length integral it produces has no elementary antiderivative, and the functions invented for it are named after this problem

The integrand is discontinuous

The answer is: The arc length integral it produces has no elementary antiderivative, and the functions invented for it are named after this problem

1.[3p] Find the area enclosed between y=x+2 and y=x2.

2.[2p] Find the area enclosed between y=2x and y=x2.

3.[3p] Revolving y=x from x=0 to x=4 about the x-axis gives what volume, to two decimal places?

4.[2p] Integrating discs to get the volume of a cone gives a factor of one third. Where does it come from?

5.[2p] How much work, in joules, stretches a spring of stiffness 200 N/m by 0.3 m?

6.[3p] What is the escape velocity from the Earth's surface, in km/s, to one decimal place?

7.[2p] The improper integral ∫1∞dxx converges.

8.[3p] A 2 m rod has density ρ(x)=1+x kg per metre. Where is its centre of mass, in metres from the light end?

9.[2p] Why is the perimeter of an ellipse a standard example of the limits of integration technique?

1.[3p]
Find the area enclosed between $y = x + 2$ and $y = x^{2}$ .

2.[2p]
Find the area enclosed between $y = 2 x$ and $y = x^{2}$ .

3.[3p]
Revolving $y = \sqrt{x}$ from $x = 0$ to $x = 4$ about the $x$ -axis gives what volume, to two decimal places?

4.[2p]
Integrating discs to get the volume of a cone gives a factor of one third. Where does it come from?

5.[2p]
How much work, in joules, stretches a spring of stiffness 200 N/m by 0.3 m?

6.[3p]
What is the escape velocity from the Earth's surface, in km/s, to one decimal place?

7.[2p]
The improper integral $\int_{1}^{\infty} \frac{d x}{x}$ converges.

8.[3p]
A 2 m rod has density $ρ (x) = 1 + x$ kg per metre. Where is its centre of mass, in metres from the light end?

9.[2p]
Why is the perimeter of an ellipse a standard example of the limits of integration technique?