Answer
This is a continuous income stream with a rate of flow of $50000$ per year, so $R(t)=50000$. Also, $k=0.09$. The capital value is given by:
$$
\begin{aligned} \int_{0}^{\infty} R(t) e^{k t} d t &=\int_{0}^{\infty} 50,000 e^{-0.09 t} d t\\
&=\lim _{b \rightarrow \infty}\int_{0}^{b} 50,000 e^{-0.09 t} d t\\
&\approx \$ 555,5556 \end{aligned}
$$
So, the capital value is $\approx \$ 555,5556$
Work Step by Step
This is a continuous income stream with a rate of flow of $50000$ per year, so $R(t)=50000$. Also, $k=0.09$. The capital value is given by
$$
\begin{aligned} \int_{0}^{\infty} R(t) e^{k t} d t &=\int_{0}^{\infty} 50,000 e^{-0.09 t} d t\\
&=\left.\lim _{b \rightarrow \infty}\left(\frac{50,000 e^{-0.09 t}}{-0.09}\right)\right|_{0} ^{b} \\
&=\lim _{b \rightarrow \infty}\left[\frac{50,000}{-0.09} e^{-0.09(b)}+\frac{50,000}{0.09}(1)\right] \\
&=\lim _{b \rightarrow \infty}\left[\frac{-50,000}{0.09 e^{0.09 b}}+\frac{50,000}{0.09}\right] \\
&=\lim _{b \rightarrow \infty}\left[\frac{-50,000}{0.09 e^{0.09 b}}\right]+\frac{50,000}{0.09} \\
&=(0)+\frac{50,000}{0.09} \\
&\approx \$ 555,5556 \end{aligned}
$$
So, the capital value is $\approx \$ 555,5556$
