Find the area between y = e^x and y = e^-x and x = 1?

Solution:

Given y = e^x and y = e^-x and x = 1

In order to find area, we need to solve the two given equation

Since both equations are equal to ‘y’. We can equate them to get x value

e^x = e^x

e^x - e^x = 0

e^x - 1/e^x = 0

e^2x - 1/e^x = 0

e^2x - 1 = 0

e^2x = 1

Apply ‘ln’ on both sides

2xlne = ln1

2x = 0

x = 0

The horizontal line is the x-axis, the vertical line is the y-axis

The orange curve is y = e^x

The blue curve is y = e^-x

The green line is x = 1

So, we determine the area of y = e^x in the interval 0 ≤ x ≤ 1 and then subtract the area of y = e^-x in the interval 0 ≤ x ≤ 1

Area = ∫\(_ 0^1\) (e^x - e^-x)dx

Area = e^x + e^-x | \(_ 0^1\)

Area = e¹ + 1/e - (1 + 1)

Area = -2 + 1/e + e

Area = -2 + 2.71 + 1/(2.71)

Area = 1.086 sq units

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Find the area between y = e^x and y = e^-x and x = 1?

Summary:

The area between y = e^x and y = e^-x and x = 1 is 1.086 sq.units