Lesson test: Triple integrals

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Triple integrals

1.[2p]
Find the volume of the tetrahedron in the first octant under the plane $2 x + y + 4 z = 8$ . Give the answer to two decimal places.

CorrectNot quite: 10.67

2.[2p]
Which iterated integral gives the volume of the tetrahedron in the first octant under the plane $x + y + z = 1$ ?

$\int_{0}^{1} \int_{0}^{1 - x} \int_{0}^{1 - x - y} d z d y d x$

Correct

$\int_{0}^{1} \int_{0}^{1} \int_{0}^{1 - x - y} d z d y d x$

The answer is: $\int_0^1 \int_0^{1 - x} \int_0^{1 - x - y} dz\,dy\,dx$

$\int_{0}^{1} \int_{0}^{1 - y} \int_{0}^{1 - x - y} d z d y d x$

The answer is: $\int_0^1 \int_0^{1 - x} \int_0^{1 - x - y} dz\,dy\,dx$

$\int_{0}^{1} \int_{0}^{1 - x} \int_{0}^{1 - y} d z d y d x$

The answer is: $\int_0^1 \int_0^{1 - x} \int_0^{1 - x - y} dz\,dy\,dx$

3.[2p]
Using cylindrical coordinates, find the volume of the solid inside the cylinder $x^{2} + y^{2} = 4$ , above the plane $z = 0$ and below the paraboloid $z = 9 - x^{2} - y^{2}$ . Give the answer to two decimal places.

CorrectNot quite: 87.96

4.[2p]
Match each coordinate system to its volume or area element.

Cartesian coordinates in space
Polar coordinates in the plane
Cylindrical coordinates
Spherical coordinates

$ρ^{2} \sin φ d ρ d φ d θ$
$d x d y d z$
$r d r d θ$
$r d r d θ d z$

Show the answer

Cartesian coordinates in space: $d x d y d z$ Polar coordinates in the plane: $r d r d θ$ Cylindrical coordinates: $r d r d θ d z$ Spherical coordinates: $ρ^{2} \sin φ d ρ d φ d θ$

5.[2p]
Find the volume of the ice cream cone cut from the ball $ρ \leq 2$ by the cone $φ \leq \frac{π}{4}$ , with $φ$ measured from the positive $z$ axis. Give the answer to two decimal places.

CorrectNot quite: 4.91

6.[3p]
A ball of radius $2$ m has density $δ = 3 - ρ$ kilograms per cubic metre, where $ρ$ is the distance from its centre. What is its mass in kilograms, to one decimal place?

CorrectNot quite: 50.3

7.[2p]
A uniform solid hemisphere of radius $12$ cm rests on its flat face. How high above the base, in centimetres, is its centre of mass?

CorrectNot quite: 4.5

8.[2p]
A uniform solid sphere has mass $5$ kg and radius $0.1$ m. What is its moment of inertia about a diameter, in kilogram square metres?

CorrectNot quite: 0.02

9.[3p]
Which of these statements are true?

Select all that apply

The ball $x^{2} + y^{2} + z^{2} \leq a^{2}$ is described by constant limits in spherical coordinates

Correct

The factor $\sin φ$ in the spherical volume element reflects that circles of latitude shrink towards the poles

Correct

In an iterated triple integral, the outermost limits may depend on the innermost variable

The answer is: The ball $x^2 + y^2 + z^2 \le a^2$ is described by constant limits in spherical coordinates, The factor $\sin\varphi$ in the spherical volume element reflects that circles of latitude shrink towards the poles, Doubling a constant density doubles the mass but leaves the centre of mass where it was

The spherical volume element is $ρ \sin φ d ρ d φ d θ$

Doubling a constant density doubles the mass but leaves the centre of mass where it was

Correct

A tetrahedron cut from the first octant by a plane fills one third of the box that contains it

1.[2p] Find the volume of the tetrahedron in the first octant under the plane 2x+y+4z=8. Give the answer to two decimal places.

2.[2p] Which iterated integral gives the volume of the tetrahedron in the first octant under the plane x+y+z=1?

3.[2p] Using cylindrical coordinates, find the volume of the solid inside the cylinder x2+y2=4, above the plane z=0 and below the paraboloid z=9-x2-y2. Give the answer to two decimal places.

4.[2p] Match each coordinate system to its volume or area element.

5.[2p] Find the volume of the ice cream cone cut from the ball ρ≤2 by the cone φ≤π4, with φ measured from the positive z axis. Give the answer to two decimal places.

6.[3p] A ball of radius 2 m has density δ=3-ρ kilograms per cubic metre, where ρ is the distance from its centre. What is its mass in kilograms, to one decimal place?

7.[2p] A uniform solid hemisphere of radius 12 cm rests on its flat face. How high above the base, in centimetres, is its centre of mass?

8.[2p] A uniform solid sphere has mass 5 kg and radius 0.1 m. What is its moment of inertia about a diameter, in kilogram square metres?

9.[3p] Which of these statements are true?

1.[2p]
Find the volume of the tetrahedron in the first octant under the plane $2 x + y + 4 z = 8$ . Give the answer to two decimal places.

2.[2p]
Which iterated integral gives the volume of the tetrahedron in the first octant under the plane $x + y + z = 1$ ?

3.[2p]
Using cylindrical coordinates, find the volume of the solid inside the cylinder $x^{2} + y^{2} = 4$ , above the plane $z = 0$ and below the paraboloid $z = 9 - x^{2} - y^{2}$ . Give the answer to two decimal places.

4.[2p]
Match each coordinate system to its volume or area element.

5.[2p]
Find the volume of the ice cream cone cut from the ball $ρ \leq 2$ by the cone $φ \leq \frac{π}{4}$ , with $φ$ measured from the positive $z$ axis. Give the answer to two decimal places.

6.[3p]
A ball of radius $2$ m has density $δ = 3 - ρ$ kilograms per cubic metre, where $ρ$ is the distance from its centre. What is its mass in kilograms, to one decimal place?

7.[2p]
A uniform solid hemisphere of radius $12$ cm rests on its flat face. How high above the base, in centimetres, is its centre of mass?

8.[2p]
A uniform solid sphere has mass $5$ kg and radius $0.1$ m. What is its moment of inertia about a diameter, in kilogram square metres?

9.[3p]
Which of these statements are true?