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 determine how many textbooks will be sold in the first three years, we need to evaluate the definite integral of the sales function S(t) over the interval [0, 3].
The integral of S(t) with respect to t can be calculated using the technique of integration by parts. The formula for integration by parts is:
∫u dv = uv - ∫v du
We need to choose u and dv such that applying the integration by parts formula simplifies the integral. In this case, let's choose:
u = t
dv = 30,000 e^(-1.5t) dt
Now, we need to find du and v.
Differentiating u with respect to t, we get:
du = dt
Integrating dv, we find:
v = ∫30,000 e^(-1.5t) dt
To evaluate v, we can use the formula for integrating exponential functions:
∫e^ax dx = (1/a) e^ax + C
In this case, a = -1.5, so:
v = (30,000 / -1.5) e^(-1.5t) + C
Now, we can apply the formula for integration by parts:
∫t (30,000 e^(-1.5t) dt) = t * [(30,000 / -1.5) e^(-1.5t)] - ∫[(30,000 / -1.5) e^(-1.5t) dt]
Simplifying this expression, we have:
∫t (30,000 e^(-1.5t) dt) = (30,000 / -1.5) te^(-1.5t) - ∫(30,000 / -1.5) e^(-1.5t) dt
We can now evaluate the integral of (30,000 / -1.5) e^(-1.5t) dt:
∫(30,000 / -1.5) e^(-1.5t) dt = (30,000 / -1.5) * [(1 / (-1.5)) e^(-1.5t) + C]
Simplifying further, we get:
∫(30,000 / -1.5) e^(-1.5t) dt = -20,000 e^(-1.5t) + C
Substituting the evaluated integral back into the original expression, we have:
∫t (30,000 e^(-1.5t) dt) = (30,000 / -1.5) te^(-1.5t) - (-20,000 e^(-1.5t) + C)
Now, we can evaluate the definite integral over the interval [0, 3]:
∫[0, 3] (30,000 te^(-1.5t) dt) = [(30,000 / -1.5) te^(-1.5t) - (-20,000 e^(-1.5t))] evaluated from 0 to 3
Using the evaluated values, we have:
[(-20,000 e^(-1.5 * 3)) - (30,000/ -1.5 * e^(-1.5 * 3))] - [(0 - 0)]
Simplifying this expression will give you the number of textbooks sold in the first three years of the edition.