Chapter 5 | Integration
597
Integrate the expression in u and then substitute the original expression in x back into the u integral: 1 2 ∫ e u du = 1 2 e u + C = 1 2 e 2 x 3 + C .
Evaluate the indefinite integral ∫ 2 x 3 e x 4 dx .
5.33
As mentioned at the beginning of this section, exponential functions are used in many real-life applications. The number e is often associated with compounded or accelerating growth, as we have seen in earlier sections about the derivative. Although the derivative represents a rate of change or a growth rate, the integral represents the total change or the total growth. Let’s look at an example in which integration of an exponential function solves a common business application. A price–demand function tells us the relationship between the quantity of a product demanded and the price of the product. In general, price decreases as quantity demanded increases. The marginal price–demand function is the derivative of the price–demand function and it tells us how fast the price changes at a given level of production. These functions are used in business to determine the price–elasticity of demand, and to help companies determine whether changing production levels would be profitable. Example 5.40 Finding a Price–Demand Equation Find the price–demand equation for a particular brand of toothpaste at a supermarket chain when the demand is 50 tubes per week at $2.35 per tube, given that the marginal price—demand function, p ′( x ), for x number of tubes per week, is given as p '( x ) =−0.015 e −0.01 x . If the supermarket chain sells 100 tubes per week, what price should it set? Solution To find the price–demand equation, integrate the marginal price–demand function. First find the antiderivative, then look at the particulars. Thus, p ( x ) = ∫ −0.015 e −0.01 x dx =−0.015 ∫ e −0.01 x dx . Using substitution, let u =−0.01 x and du =−0.01 dx . Then, divide both sides of the du equation by −0.01. This gives −0.015 −0.01 ∫ e u du =1.5 ∫ e u du =1.5 e u + C =1.5 e −0.01 x + C . The next step is to solve for C . We know that when the price is $2.35 per tube, the demand is 50 tubes per week. This means
Made with FlippingBook - professional solution for displaying marketing and sales documents online