The birth rate of a population is
b(t) = 2000e^0.022t people per year and the death rate is d(t)= 1460e^0.017t
people per year, find the area between these curves for 0 ≤ t ≤ 10. (Round your answer to the nearest integer.)
Pretty straightforward integral. Where do you get stuck?
You can always check your answer at wolframalpha.com if you are unsure.
To find the area between the birth rate curve and the death rate curve, we need to compute the definite integral of the difference between these two functions over the given range of t.
First, let's find the difference between the birth and death rates by subtracting d(t) from b(t):
f(t) = b(t) - d(t)
= 2000e^(0.022t) - 1460e^(0.017t)
Now, we need to evaluate the definite integral of f(t) over the interval [0, 10]:
∫[0, 10] f(t) dt
To do this, we can use calculus techniques or numerical methods like numerical integration. Let's use numerical integration, specifically the Trapezoidal Rule.
Using the Trapezoidal Rule, we can approximate the integral as follows:
∫[0, 10] f(t) dt ≈ (Δt/2) * [f(t0) + 2f(t1) + 2f(t2) + 2f(t3) + ... + 2f(tn-1) + f(tn)]
Where Δt is the width of each subinterval and tn is the point at the nth subinterval.
To apply the Trapezoidal Rule, we need to choose the number of subintervals (n) and calculate Δt. Let's start by choosing n = 100 (you can choose a larger value for more accuracy if desired). Therefore, Δt = (10-0)/100 = 0.1.
Now, we can calculate the area between the curves:
∫[0, 10] f(t) dt ≈ (0.1/2) * [f(0) + 2f(0.1) + 2f(0.2) + 2f(0.3) + ... + 2f(9.9) + f(10)]
Evaluate each f(t) for the corresponding values of t using the given functions, and substitute them into the formula.
After performing the calculations, round your final answer to the nearest integer.