Calculus 1 : How to find integral expressions

Study concepts, example questions & explanations for Calculus 1

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Example Questions

Example Question #121 : Integral Expressions

Determine the integral expression of the volume of a cube , using its surface area  and length .

Possible Answers:

Correct answer:

Explanation:

First, recall that surface area of a cube is related to the area of one its faces  by

.

Since volume of a cube is related to the area of one of its faces  multiplied by , we can write volume in terms of  and  as

Since we want an integral expression, we can define volume as the definite integral of area. 

We also know that for definite integrals,

, where

In our case, 

Since 

 

Example Question #121 : How To Find Integral Expressions

Possible Answers:

Correct answer:

Explanation:

To integrate this, first make it a little easier on yourself by chopping the whole expression into three separate terms. (When you have one denominator, you can do that!).

It then looks like this:

.

Then, simplify:

.

Now, integrate each term separately. Remember to raise each term's exponent by 1 and also put that result on the denominator.

Therefore, the integral is: 

.

Since there's no particular solution, it is an indefinite integral and you have to put "C" at the end:

.

Example Question #123 : Integral Expressions

Possible Answers:

Correct answer:

Explanation:

To integrate this expression, you must use "U" substitution.

First, assign your u.

.

Then, find the derivative of u:

.

Since you don't have an 8 in the original problem, you must divide out that 8 so you get

.

Now you can plug in! I'd keep the coefficient outside of the integral sign:

.

Focus on just integrating u; rewrite as .

Then, add 1 to the exponent and put that result on the denominator:

.

That becomes . We then remember to multiply that expression by the that we took out previously. We then get .

We then sub our expression back in for u and remember to take on a C because it is an indefinite integral.

Our final answer is:

.

Example Question #122 : How To Find Integral Expressions

Possible Answers:

Correct answer:

Explanation:

To integrate this expression, I'd first rewrite it so that all the terms have fractional exponents--they make it easier to integrate

.

Then, integrate each term, remembering to raise the exponent by 1 and then also putting that resulting value in the denominator:

.

Then, simplify all of that and remember to add a "C" at the end because it is an indefinite integral:

.

Example Question #123 : How To Find Integral Expressions

Possible Answers:

Correct answer:

Explanation:

To integrate this expression, focus on each term separately. To integrate , set the 4 outside while integrating t. Remember to increase the exponent by 1 and then put that result in the denominator:

.

Simplify and you get:

.

Move on to the next term remembering these steps.

becomes  when integrated.

t becomes when integrated.

Now, link all of these together and add a "C" at the end because it is an indefinite integral:

.

Example Question #124 : How To Find Integral Expressions

Possible Answers:

Correct answer:

Explanation:

To make this integration a little easier, chop it up into two terms since you only have 1 denominator:

.

Then, integrate each one separately. Remember, when integrating, raise the exponent of a term by 1 and then put that result on the denominator.

.

.

Link those together and add a "C" at the end because it is an indefinite integral:

Example Question #125 : How To Find Integral Expressions

Find the integral of the following function:

Possible Answers:

Correct answer:

Explanation:

The integral was performed using the following rules:

.

The original function can be rewritten as follows.

Now apply the above rules to the two new integrals to find the final solution.

Remember to add the constant of integration C since this is an indefinite integral.

Example Question #126 : How To Find Integral Expressions

Which integral represents the area of the space confined by the functions , and 

Possible Answers:

Correct answer:

Explanation:

First, note that these function intersect at (0,0) and (1,1), so the limits of integration should be from 0 to 1. This is because  when .

Then, to find the area, we must take the integral over the one on top minus the one on the bottom.

In this case x is on top, so the integral is 

.

Example Question #127 : How To Find Integral Expressions

Solve the indefinite integral.

Possible Answers:

None of these

Correct answer:

Explanation:

To solve this integral, we must use integration by parts. Integration by parts states that . We can set u and dv and then find du and v by knowing tthat the derivative of  is  and the derivative of  is itself.

 

So the solution is 

We must add the C because the integral is indefnite.

 

Example Question #128 : How To Find Integral Expressions

Solve the indefinite integral

Possible Answers:

None of these

Correct answer:

Explanation:

An integral is the opposite of a derivative so we must use reverse derivative rules. The integral of  is . Using that we can solve this integral.

We need to add C because the integral is indefinite.

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