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A demand curve is given by $ p = \frac{450}{(x + 8)} $. Find the consumer surplus when the selling price is $10.

$$450 \ln \left(\frac{45}{8}\right)-370 \approx \$ 407.25$$

Applications of Integration

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So in this problem, we're given a demand curve. P is equal to 450 divided by the quantity X plus eight and were asked to find consumer surplus when the price So consumer surplus on this graph is going to be this area in green Now, To find it, we have to integrate this demand curve up to the value it corresponds it 10 and then subtract out the area of this box. So to begin first, we need a value where I have placed that red dot. So to get it, we're going to take 10 which is a value of P and said it equal to 450 over the quantity expose. So if we multiply both sides by expose, we're going to get 10 eggs. Course, can you equal to 450 and then subtracting 80 from both sides. We get 10 x is equal to 370 and x his equal 2 37 So 37 units is what we have for X now to get the green area, which is our consumer surplus that is going to be equal to the integral from zero 2 37 of 450 over X plus eight d X and then subtracting off 10 times 37 which is going to be the area of a rectangle in red. So this integral right here, we'll evaluate 2 450 And since we have a X to the power of one term and the denominator, we'll get the natural log of X plus eight evaluated at zero and 37. In most cases, you have to put absolute value bars around the argument for the natural log. But here, if you plug in zero, you get positive. Eight. If you book in 37 you get positive. 45 and so you will never go negative. So you can just put parentheses in this situation, and then we're going to subtract off 370. So we get for 50 times dear natural log of 45 minus the natural log of eight minus our 370. So 450 just to simplify for the exact form, slightly weaken right as 45 over eight and then again minus 3 70 So this would be the answer in exact form, and then to approximate this first term 450 times. The natural log of 45 of her eight is approximately 777 0.25 And so subtracting off we get 407 dollars and 25 cents for consumer surplus.