Calculus 3 : Integration

Study concepts, example questions & explanations for Calculus 3

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Example Questions

Example Question #11 : Integration

A particle's velocity in two dimensions is described by the functions:

If the particle has an initial position of , what will its position be at time ?

Possible Answers:

Correct answer:

Explanation:

Position can be found by integrating velocity with respect to time:

For velocities:

The position functions are:

These constants of integration can be found by using the given initial conditions:

Which gives the definite integrals:

Example Question #12 : Integration

Calculate antiderivative of 

Possible Answers:

Correct answer:

Explanation:

We can calculate the antiderivative 

by using -substitution. We set , so we get , so the integral becomes

So we just plug  to get

Example Question #111 : Calculus Review

Calculate antiderivative of 

Possible Answers:

Correct answer:

Explanation:

We can calculate the antiderivative 

by using the power rule for antiderivatives: 

In this case , so we have

.

Example Question #14 : Integration

The physical interpretation of the integral of a function , denoted by , is what?

Possible Answers:

No physical significance.

The average value of the tangent lines on .

The points of inflection of the function .

Area under the function .

Correct answer:

Area under the function .

Explanation:

By definition, the integral of a fucntion  is the summation of an infinite number of small areas, thus giving the total area of the function with respect to the axis of integration.

Example Question #11 : Integration

Calculate the integral of the function  given below.

Possible Answers:

Correct answer:

Explanation:

We have a seperable integral, meaning each term can be integrated independently, written as

.

The first term is a power-rule integral, while the second is a trigonometric intergral.  One should recall

and 

Putting these facts together leads us to the final answer of

.

Example Question #161 : How To Find Position

If the acceleration function of an object is , what is the position of the object at ? Assume the initial velocity and position is zero.

Possible Answers:

Correct answer:

Explanation:

To find the position function from the acceleration function, integrate  twice.

When integrating, remember to increase the exponent of the variable by one and then divide the term by the new exponent. Do this for each term.

Solve for .

Example Question #15 : Integration

Evaluate .

Possible Answers:

None of the other answers

Correct answer:

None of the other answers

Explanation:

The correct answer is .

 

We can evaluate this integral using -substitution.

Let , then , hence we have

(Don't forget to change the bounds of integration)

.

 

 

Example Question #12 : Integration

Evaluate , where is any constant.

Possible Answers:

None of the other answers

Correct answer:

Explanation:

Since is a constant, so is , therefore we can factor it out of the integral.

 

Example Question #18 : Integration

Calculate .

Possible Answers:

Correct answer:

Explanation:

This integral can be found using u-substitution.  Consider

.  This means we can rewrite our integral as

, by definition of the integral of an exponential.

Example Question #17 : Integration

Calculate 

Possible Answers:

Can not be determined.

Correct answer:

Explanation:

This integral can be done using integration by parts.  Consider

.

Choose  and .

Using the definition of integration by parts,

,

.

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