# Worksheet: Integral Curves of Vector Fields

In this worksheet, we will practice finding the integral curve of a vector field.

Q1:

The figures show the vector field , together with several of its flows. Suppose we know that for some numbers the integral curves are such that and are linear combinations of some . What are the values of ?

• A and 2
• B and 3
• C and 3
• D and 2
• E and 2

What are the parametric equations of the integral curve that is at when ?

• A
• B
• C
• D
• E

What are the parametric equations of the integral curve that is at when ?

• A
• B
• C
• D
• E

What are the parametric equations of the integral curve that is at when ?

• A
• B
• C
• D
• E

As and as along an integral curve, the secant between and approaches one of the lines and shown. What are the slopes of these two lines?

• Aslope of , slope of
• Bslope of , slope of
• Cslope of , slope of
• Dslope of , slope of
• Eslope of , slope of

Q2:

The figures show the vector field , together with several of its flows. Suppose we know that, for some numbers , the integral curves and are such that and are linear combinations of some . What are the values of ?

• A and 2
• B and
• C and
• D and 2
• E and

What are the parametric equations of the integral curve that is at when ?

• A,
• B,
• C,
• D,
• E,

What are the parametric equations of the integral curve that is at when ?

• A,
• B,
• C,
• D,
• E,

What are the parametric equations of the integral curve that is at when ?

• A,
• B,
• C,
• D,
• E,

Using the fact that , find a Cartesian equation satisfied by the points of the integral curve that is at when . You need not simplify your expression.

• A
• B
• C
• D
• E

As and along an integral curve, the secant between and approaches one of the lines and shown. What are the slopes of these two lines?

• ASlope of , slope of
• BSlope of , slope of
• CSlope of , slope of
• DSlope of , slope of
• ESlope of , slope of

Q3:

Consider the parametric curve , with constants and . The figure shows the case and for . Find a vector field such that the curve and is its integral curve.

• A
• B
• C
• D
• E

Find a linear second-order differential equation satisfied by .

• A
• B
• C
• D
• E

You can check that is also a solution to this differential equation and therefore any for constants and . Using the vector field, determine the corresponding function so that and is an integral curve.

• A
• B
• C
• D
• E

For the case and , find parametric equations for the integral curve that starts at the point when .

• A,
• B,
• C,
• D,
• E,

Q4:

The figures show the vector field , together with several of its flows. Suppose we know that, for some numbers , the integral curves and are such that and are linear combinations of some . What are the values of ?

In this case, where is a repeated root, linear combinations of and are used. Hence, find the parametric equations of the integral curve that is at when .

• A,
• B,
• C,
• D,
• E,

What are the parametric equations of the integral curve that is at when ?

• A,
• B,
• C,
• D,
• E,

What are the parametric equations of the integral curve that is at when ?

• A,
• B,
• C,
• D,
• E,

As and along an integral curve, the secant between (0, 0) and approaches the dashed line shown. What is the slope of this line?

• A
• B
• C
• D
• E

Q5:

If parameterizes an integral curve of the vector field , then . This means is a linear combination of and .

Find the -parameter function for the integral curve to this vector field that starts at the point .

• A
• B
• C
• D
• E

Find the Cartesian equation of the integral curve determined above.

Hint: It is a hyperbola.

• A
• B
• C
• D
• E

Find the Cartesian equation of the integral curve to this vector field that starts at the point .

• A
• B
• C
• D
• E

Q6:

An integral curve (or flow) of a vector field is a parametric curve with for every where and are defined.

By solving the equations and , find an integral curve for the vector field that also satisfies .

• A
• B
• C
• D
• E

Consider the vector field . Find the Cartesian equation of the vector field’s integral curve which is at the point when .

• A
• B
• C
• D
• E

Find the Cartesian equation of the integral curve to that starts at the point (2, 2).

• A
• B
• C
• D
• E

Find the Cartesian equation of the integral curve to that starts at the point (2, 2).

• A
• B
• C
• D
• E

Find the parametric equations of the integral curve to that starts at the point (0, 2).

• A
• B
• C
• D
• E

Do the integral curves of the vector fields and starting at (0, 2) describe the same set in for ?

• Ayes
• Bno

Do the integral curves of the vector fields and starting at (2,2) describe the same set in for ?

• Ayes
• Bno

The integral curves to and that start at both lie inside the curve but go in opposite directions. Determine the parametric equations that integrate the vector field and start at .

• A
• B
• C
• D

Do the integral curves of the vector fields and starting at (0, 2) describe the same set in for ?

• Ano
• Byes