Question

Let R be the region between the graph of y=e^-2x and the x-axis for x greater than or = to 3. The area of R is?

Answer 1

 When you substitute a 3 you get 1/2e∧(-6) and when you substitute a t it just goes to 0
 
Answer
1/2e(-6)

Answer 2

Answer:  The required area of region R is $\dfrac{1}{2e^{6}}.$

Step-by-step explanation:  Given that 'R' is the region between the graph of $y=e^{-2x}$ and X-axis for $x\geq 3.$

We are given to find the area of the region 'R'.

To find the area of region 'R', we need to integrate the given function from x = 3 to x = ∞.

The area of region 'R' can be calculated as follows:

$A\\\\\\=\int_{x=3}^{\infty}ydx\\\\\\=\int_{x=3}^{\infty}e^{-2x}dx\\\\\\=\left(\dfrac{e^{-2x}}{-2}\right)_{x=3}^{\infty}\\\\\\=-\dfrac{1}{2}(e^{-\infty}-e^{-2\times3})\\\\\\=-\dfrac{1}{2}\left(\dfrac{1}{e^\infty}-\dfrac{1}{e^6}\right)\\\\\\=-\dfrac{1}{2}\left(\dfrac{1}{\infty}-\dfrac{1}{e^6}\right)\\\\\\=-\dfrac{1}{2}\left(0-\dfrac{1}{e^6}\right)\\\\\\=\dfrac{1}{2e^6}.$

Thus, the required area of the region 'R' is $\dfrac{1}{2e^{6}}.$