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The supply function $ p_s (x) $ for a commodity gives the relation between the selling price and the number of units that manufacturers will produce at that price. For a higher price, manufacturers will produce more units, so $ p_s $ is an increasing function of $ x $. Let $ X $ be the amount of the commodity currently produced and let $ P = p_s (X) $ be the current price. Some producers would be willing to make and sell the commodity for a lower selling price and are therefore receiving more than their minimal price. The excess is called the producer surplus. An argument similar to that for consumer surplus shows that the surplus is given by the integral

$$ \int_0^X [P - p_s (x)]\ dx $$

Calculate the producer surplus for the supply function $ p_s (x) = 3 + 0.01x^2 $ at the sales level $ X = 10 $. Illustrate by drawing the supply curve and identifying the producer surplus as an area.

$\frac{20}{3}$

Applications of Integration

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Okay, So our problem gave us a supply curve piece of S of X and asked us to find producer surplus when there are 10 units. So first. So we're clear. Producer surplus is the area of shaded in green. Find it. First, we're going to find the corresponding price to 10 units. Then we're going to take this big area that I've shaded in blue and subtract off that red area. So to begin first, to find the corresponding price, we simply evaluate piece of s of 10 which is three plus 0.1 x. In this case, you plug in, get 10 squared now 0.1 is one over 110 squared is 100. So together those will evaluate toe one and three plus one is equal to four. So that's our price. Now what we're going to dio to get our value for producer surplus is equal to the blue area which is just the area of a rectangle with side lengths four and 10 and then we're going to subtract off the red area, which is the integral from zero 2 10 of our supply function, which is three plus 0.1 x squared DX that is going to be equal to 40 minus in red three x from 0 to 10 plus 0.1 over three x cubed from 0 to 10 now, since these terms are just in terms of X, when you plug in zero, it will zero out. So we only need to plug in 10 teach. And when we do that next step we get 40 minus three times 10 is 30 and then here 10 cubed is 1000 times 1 1/100 is going to be 10 over three now to put everything in terms of a common denominator, we're going to get 40 times three is 1 20 minus 30 times three is 90. That minus the 10 and all of that divided by three and 90 minus 10 minus 90 minus 10 is minus 101 20 minus 100 is simply our final answer of 20 over three