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 , 

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:

.

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:

.

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:

.

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:

.

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:

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.

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 

.

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.

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.