# Calculus definite integrals declining word problem

The marriage rate in the United States has been declining recently, with about $2.1e^{-0.034t}$ million marriages per year where $t$ is the number of years since 2008. Assuming that this rate continues, find the total number of marriages in the United Stats from 2008 to 2018.

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Hints:

• Write $2.1\cdot \mathrm e^{-0.034t}$ as $a\cdot b^t$ for some $a$ and $b$.
• If 2008 is year $0$, then 2018 is year $10$.
• $\sum\limits_{t=0}^{10}a\cdot b^t=a\cdot\dfrac{1-b^{11}}{1-b}$.
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This looks like a homework problem.

Hint:

1. Integration of exponential functions: $$\int a e^{bt} dt = a \int e^{bt} dt = \frac{a}{b} e^{bt} + \text{const}.$$

2. Definite integrals: $$\int_{x}^{y} f(t) dt = F(y) - F(x), \text{ where } F'(t)= f(t).$$

Can you take it from here? Can you fill in the values of $a, b, x$ and $y$?

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