Question

What is the approximate area of the curve y = -e ^ (- 4x) between the x axis in the range (0, + ∞) ?

Answer 1

$$\begin{aligned} &\text{ The area of the curve y between the x axis given by }\\[1 em] \int_{0}^{\infty} - e^{-4 x} d x&=\lim _{b \rightarrow \infty} \int_{0}^{b} - e^{-4 x} d x \\[1 em] &=\frac{1}{4}\lim _{b \rightarrow \infty}\left[-e^{-4 x}\right]_{0}^{b} \\[1 em] &=\frac{1}{4}\lim _{b \rightarrow \infty}\left[- e^{-4b}-\left(- e^{0}\right)\right] \\[1 em] & \text { Note that } \ e^\infty =\infty \\[1 em] &\Rightarrow \lim _{b \rightarrow \infty}- \frac{1}{e^{4b}}= -\frac{1}{e^{\infty}}=-\frac{1}{\infty}=0\\[1 em] &\Rightarrow\frac{1}{4}\lim _{b \rightarrow \infty}\left[- \frac{1}{e^{4b}} -\left(- e^{0}\right)\right] =\frac{1}{4}(0-(-1))=\frac{1}{4} \end{aligned}$$