A new college textbook edition typically generates most of its sales in the year of its publication. Sales drop off in subsequent years as a result of competition from the used book market. Suppose that the annual sales of a particular textbook may be modeled by
S(t) = 30,000 te−1.5t, textbooks, where t is the number of years since the edition was published.
Use integration by parts to determine how many textbooks will be sold in the first three years of the edition?
To find out how many textbooks will be sold in the first three years of the edition, we need to calculate the definite integral of the given sales function over the interval [0, 3].
The formula for integration by parts is ∫u dv = uv - ∫v du, where u and v are functions of the variable t. In this case, we can choose:
u = t
du = dt
dv = 30,000 e^(-1.5t) dt
v = ∫dv = -20,000 e^(-1.5t)
Using the integration by parts formula, we have:
∫(t * 30,000 e^(-1.5t)) dt = -20,000 t e^(-1.5t) - ∫(-20,000 e^(-1.5t) dt)
Now let's calculate the definite integral over the interval [0, 3]:
∫[0,3] (t * 30,000 e^(-1.5t)) dt = [-20,000 t e^(-1.5t) - ∫(-20,000 e^(-1.5t) dt)] evaluated from 0 to 3
Plugging in the upper and lower limits for t, we get:
[-20,000 (3) e^(-1.5(3)) - ∫[-20,000 e^(-1.5t)] evaluated from 0 to 3
Simplifying further:
[-20,000 (3) e^(-4.5) - (∫[-20,000 e^(-1.5t)] evaluated from 0 to 3)
Now we need to evaluate the indefinite integral term. The integral of -20,000 e^(-1.5t) with respect to t is:
∫[-20,000 e^(-1.5t)] dt = (-20,000 / -1.5) e^(-1.5t) = 13,333.333 e^(-1.5t)
Now let's evaluate the integral over the interval [0, 3]:
[-20,000 (3) e^(-4.5) - (13,333.333 e^(-1.5t)] evaluated from 0 to 3
[-20,000 (3) e^(-4.5) - (13,333.333 e^(-1.5(3))] - [-20,000 (0) e^(-4.5) - (13,333.333 e^(-1.5(0))]
Simplifying further:
[-20,000 (3) e^(-4.5) - 13,333.333 e^(-4.5)] - [0 - 13,333.333]
Finally, we have:
-60,000 e^(-4.5) - 13,333.333 e^(-4.5) + 13,333.333
Calculating this expression, we find that approximately 7,112 textbooks will be sold in the first three years of the edition.