To find the area between the curves y = e^x and y = e^(-x) from x = 0 to x = 1, we first need to identify the points where these two curves intersect.
Setting e^x = e^(-x), we can rewrite this as:
1 = e^{-2x} → e^{2x} = 1 → 2x = 0 → x = 0
The curves intersect at the point (0, 1). Now, we will calculate the area between the curves from x = 0 to x = 1.
The area can be expressed as:
A = ∫ (e^x - e^(-x)) dx from 0 to 1.
Now we can calculate the integral:
Calculating the antiderivative:
∫(e^x)dx = e^x + C
∫(e^(-x))dx = -e^(-x) + C
So, we have:
∫(e^x - e^(-x)) dx = e^x + e^(-x) + C
Now we will evaluate from x = 0 to x = 1:
A = [e^x + e^(-x)] from 0 to 1
Calculating it:
A = (e^1 + e^(-1)) - (e^0 + e^0) = (e + 1/e) - (1 + 1) = e + 1/e - 2
Now, plugging in the value of e ( 2.71828), we find that:
So, the area between the curves from x = 0 to x = 1 is:
A ≈ 2.71828 + 0.367879 - 2 ≈ 1.086159
Thus, the area between the curves y = e^x and y = e^(-x) from x = 0 to x = 1 is approximately 1.086159 square units.