Calculus 1 : How to find integral expressions

Study concepts, example questions & explanations for Calculus 1

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Example Questions

Example Question #1071 : Functions

Set up the integral to find the area of a region bounded by the curve  from  to .

Possible Answers:

Correct answer:

Explanation:

In order to find the area of the region under the curve of the parabola, simply integrate from the given bounds.  Ensure that there is no negative area within the bounds by setting the parabola equal to zero.

After testing the numbers to the left and right of the critical values, the positive region is in between bounds.

Set up the integral.

Example Question #21 : Integral Expressions

Evaluate:

Possible Answers:

Correct answer:

Explanation:

We get this by applying the inverse power rule to each polynomial term. 

Therefore we get,

Since this is an indefinite integral, we need to add a constant term  at the end of the function.

Example Question #1071 : Functions

Evaluate:

Possible Answers:

Correct answer:

Explanation:

In order to evaluate the integral, we need to use the inverse power law and the integral of trig functions.

Remember that the inverse power rule is

 and the integral of sine is a negative cosine.

 

Example Question #1072 : Functions

Evaluate:

Possible Answers:

Correct answer:

Explanation:

In order to evaluate the integral, we need to use the inverse power rule. 

Remember that the inverse power rule is

 

Remember to have the constant term since this is an indefinite integral.

Example Question #25 : Integral Expressions

Evalulate:

 

Possible Answers:

Correct answer:

Explanation:

We evaluate the definite integral by using the definition of the integral for  .

 

 

Example Question #26 : Integral Expressions

A given derivative  is defined as . What is ?

Possible Answers:

None of the above

Correct answer:

Explanation:

In order to find , we must take the indefinite integral of the given derivative .

Noting that the power rule for integrals is 

 for all  and with an arbitrary constant of integration ,

we can determine that 

.

Example Question #27 : Integral Expressions

A given derivative  is defined as . What is ?

Possible Answers:

None of the above

Correct answer:

Explanation:

In order to find , we must take the indefinite integral of the given derivative .

Noting that the power rule for integrals is 

 for all  and with an arbitrary constant of integration ,

we can determine that 

.

Example Question #21 : How To Find Integral Expressions

A given derivative  is defined as . What is ?

Possible Answers:

None of the above

Correct answer:

Explanation:

In order to find , we must take the indefinite integral of the given derivative .

Noting that the power rule for integrals is 

 for all  and with an arbitrary constant of integration ,

we can determine that 

.

Example Question #21 : Writing Equations

Evaluate the indefinite integral:

Possible Answers:

Correct answer:

Explanation:

To integrate, one must use the "u-substitution" or substitution method for integration, because one cannot simply integrate

In this case, we are calling , and therefore .

Replace  with , and  with .

The integral becomes far easier:

 , and is equal to .

The rule used for this integration is

.

(The constant -3/2 remains unchanged.) Finally, replace  with  to go back to the original variable/term being integrated.

Example Question #21 : Writing Equations

Evaluate .

Possible Answers:

Correct answer:

Explanation:

Take the anti-derivative (integrate using the Power Rule : ) of the expression:

Plug in the number at the top of the integral:

Then, plug in the number at the bottom of the integral:

Lastly, subtract the top from the bottom:

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