A frozen food company is preparing to ship $2000$ units to a grocery store chain's distribution center when the chain contacts them to increase their order to $10000$ units. The frozen food company knows that when

thousand units are shipped, the marginal revenue is expressed as

thousand dollars per thousand items. Also, when

thousand units are packaged and shipped, the marginal cost function is

thousand dollars per thousand items. What is the net increase in profit for the frozen food company when the chain increased their order from $2000$ units to $10000$ units?

$($a$)$ First, we will set up the definite integral that will be used to answer the question, then continue to fill in the blanks to perform the steps to evaluate the integral.

$($b$)$ Complete the following sentence:

When $2000$ units have already been ordered, increasing the order size by

units will result in a net increase in profit of $

for the frozen food company.

$($c$)$ The finance office of the frozen food company sees that the total profit for the original order of $2000$ units was going to be $$3040.$ Now, we know that the total profit for the entire order was $

$.$ This means that the average profit was

dollars per unit $($round to the nearest cent, if necessary$).$