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Chapter 8 – Further Applications of Integration

Chapter 8 – Further Applications of Integration. 8.4 Applications to Economics and Biology. Applications to Physics and Engineering. Among the many applications of integral calculus to economics and biology, we will consider these today: Consumer Surplus Blood Flow Cardiac Output

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Chapter 8 – Further Applications of Integration

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  1. Chapter 8 – Further Applications of Integration 8.4 Applications to Economics and Biology 8.4 Applications to Economics and Biology

  2. Applications to Physics and Engineering • Among the many applications of integral calculus to economics and biology, we will consider these today: • Consumer Surplus • Blood Flow • Cardiac Output • Remember, the marginal cost function C′(x) was defined to be the derivative of the cost function. Refer to sections 3.7 and 4.7. 8.4 Applications to Economics and Biology

  3. Consumer Surplus • The graph of a typical demand function is called a demand curve. If X is the amount of the commodity that is available, then P = p(X) is the current selling price. • Divide the interval [0, X] into n subintervals of length x. • The amount saved is (savings per unit)(number of units) = [p(xi) – P] x The total savings is given by the Riemann Sum: 8.4 Applications to Economics and Biology

  4. Consumer Surplus • If we let n ∞, the Riemann sum approaches the integral which is called the consumer surplus for the commodity. • The consumer surplus represents the amount of money saved by consumers in purchasing the commodity at price P which corresponds to an amount demanded of X. 8.4 Applications to Economics and Biology

  5. Example 1 – pg. 566 #4 • The demand function for a certain commodity is Find the consumer surplus when the sales level is 300. Illustrate by drawing the demand curve and identifying the consumer surplus as an area. 8.4 Applications to Economics and Biology

  6. Blood Flow • The equation is called Poiseuille’s Law, and it shows that the flux is proportional to the 4th power of the radius of the blood vessel. • Note: F is called the flux, R is the radius of the blood vessel, l is the length of the blood vessel, P is the pressure between the ends of the blood vessel, and  is the viscosity of the blood. 8.4 Applications to Economics and Biology

  7. Cardiac Output • The cardiac output of the heart is the volume of blood pumped by the heart per unit of time, that is, the rate of flow into the aorta. The cardiac output is given by Note: F is the flow rate, A is the amount of dye known, and c(t) is the concentration of the dye at time t. 8.4 Applications to Economics and Biology

  8. Example 2 – pg. 567 #18 • After an 8-mg injection of dye, the reading of the dye concentration in mg/L at two second intervals are shown in the table. Use Simpson’s Rule to estimate the cardiac output. 8.4 Applications to Economics and Biology

  9. Book Resources • Video Examples • Example 1 – pg. 564 • Example 2 – pg. 566 • More Videos • Using Integrals to find Consumer Surplus • Wolfram Demonstrations • Consumer and Producer Surplus 8.4 Applications to Economics and Biology

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