.Find the length of the curve y=e^x+1/4 e^−x between x=0 and x=1.
To find the length of a curve between two points, you can use a method called "arc length integration." This approach involves integrating the square root of the sum of the squares of the derivatives of the curve's equation with respect to x, over the desired interval.
Let's start by finding the derivative of the curve equation, which is y = e^x + (1/4)e^(-x).
dy/dx = d/dx(e^x) + d/dx((1/4)e^(-x))
= e^x - (1/4)e^(-x)
Next, we square this derivative:
(dy/dx)^2 = (e^x - (1/4)e^(-x))^2
= e^(2x) - 2(e^x)(1/4)e^(-x) + (1/16)e^(-2x)
= e^(2x) - (1/2) + (1/16)e^(-2x)
Now we integrate the square root of this expression over the interval [0, 1]:
L = ∫[0,1] √(1 + (dy/dx)^2) dx
= ∫[0,1] √(1 + (e^(2x) - (1/2) + (1/16)e^(-2x))) dx
Since the integral is a bit involved, let's simplify the expression inside the square root first:
1 + (e^(2x) - (1/2) + (1/16)e^(-2x))
= 1 + e^(2x) - (1/2) + (1/16)e^(-2x)
= (1/2) + e^(2x) + (1/16)e^(-2x)
Now, let's integrate the square root of this expression over the interval [0, 1]:
L = ∫[0,1] √((1/2) + e^(2x) + (1/16)e^(-2x)) dx
At this point, the calculation becomes quite involved and requires numerical integration methods or specialized software. You can use numerical integration techniques like Simpson's rule or the trapezoidal rule to approximate the integral and find the length of the curve.
Alternatively, you can use software like Wolfram Alpha or a mathematical software package (such as MATLAB or Python with libraries like SciPy) to compute the integral numerically. Simply input the integral expression, specify the interval [0, 1], and the software will provide you with the approximate length of the curve.