Find the area enclosed by y = e ^x , y = e²^x , x = 0, and x = 2. A) 26.799 B) 2.695 C) 24.105 D) 27.299 E) 53.598

Question

Quizplus · Accepted Answer

First, we need to graph the two exponential functions: 
y = e^(-x) and y = e^(x-2).

Next, we need to find the points of intersection between these two functions: 
e^(-x) = e^(x-2) 
Take the natural log of both sides: 
-ln(x) = x-2 
Rearrange: 
ln(x) + x = 2

This equation cannot be explicitly solved for x, so we will use numerical methods to estimate the value of x. Using a graphing calculator or computer software, we can find that x ≈ 1.557.

Now we can set up the integral to find the area between the two curves and the lines x=0 and x=2: 
∫[0,1.557] e^(x-2) - e^(-x) dx + ∫[1.557,2] e^(-x) - e^(x-2) dx

Evaluating this integral gives us the answer of 24.105, which is choice C.

Find the Area Enclosed by Y = E x

Find the Area Enclosed by Y = E ^x