#### To determine

**To explain:**

The meaning of the area of the shaded region and use midpoint rule to estimate the value of the quantity

#### Answer

The shaded region represents the difference between the areas under R'xand C'(x). It is the total profit from 50 units produced after the first 50 units. The value of the quantity ≈50.5 thousand dollars.

#### Explanation

**1) Concept:**

*Midpoint rule ∫baf(x)dx≈∑i=1nfxi-∆x=∆x[fx1-+…+fxn-]*

where, ∆x=b-an and

*xi-=12xi-1+xi= Midpoint of [xi-1, xi]*

2) **Calculation:**

The area under R'(x) from x=50 to x=100 represents the change in revenue, and the area under C'(x) represents the change in cost.

So the shaded region represents the difference between the areas under R'xand C'(x)

Let P'x=R'x- C'(x)

The rate of increase in the profit as production levels change from 50 to 100 units is R'x- C'(x). This is marginal profit function. So the shaded area represents total profit as production levels change from 50 to 100 units.

With n=5,a=50, and b=100,*the * interval width is

*∆x=b-an*

Substitute values.

*∆x=100-505*

Simplify.

*∆x=505*

Simplify.

*∆x=10*

Use midpoint rule with ∆x=10 and n=5.

*M5=∆xR'55-C'55+R'65-C'65+R'75-C'75+R'85-C'85+R'95-C'95*

From the graph R'55=2.4, R'65=2.2, R'75=2, R'85=1.8, R'95=1.7 and

*C'55=0.85, C'65=0.9, C'75=1, C'85=1.1, C'95=1.2*

Substitute above values in M5.

*M5=102.4-0.85+2.2-0.9+2-1+1.8-1.1+1.7-1.2*

Simplify the above equation.

*M5=10(5.05)*

Simplify.

M5≈50.5 thousand dollars

Therefore, the value of the quantity is 50.5 thousand dollars.

**Conclusion:**

The shaded region represents the difference between the areas under R'xand C'(x), and the value of the quantity is 50.5 thousand dollars.