Finding the Length of the Curve y = ln(1 + x2) from x = 0 to x = 1.6
The length of a curve defined by a function y = f(x) from x = a to x = b can be calculated using the formula:
L = ∫ab √(1 + (f'(x))2) dx
Step 1: Define the function
For our case, the function is:
y = ln(1 + x2)
Step 2: Find the derivative of the function
To calculate the length, we need to find the derivative y':
Using the chain rule and the derivative of the natural logarithm, we find:
y' = d/dx [ln(1 + x2)] = (1/(1 + x2)) * (2x) = &frac2{1 + x2}x
Step 3: Square the derivative
Next, we square the derivative:
(y')2 = &left;(&frac2{2x}{1 + x2}&right;)2 = &frac4{x2}{(1 + x2)2}
Step 4: Write the formula for arc length
Now we can substitute this back into the arc length formula:
L = ∫01.6 √&left;(1 + &frac4{x2}{(1 + x2)2}&right;) dx
Step 5: Simplifying the integrand
The integrand simplifies to:
√&left;(1 + &frac4{x2}{(1 + x2)2}&right; = rac{&radic{(1 + x2)2 + 4}}{1 + x2}
Step 6: Evaluate the integral
Now, we need to evaluate the integral from 0 to 1.6:
L = ∫01.6 &frac{&radic{(1 + x2)2 + 4}}{1 + x2} dx
This integral may not have a straightforward primitive, so numerical methods like Simpson’s Rule or numerical integration could be used to approximate this integral.
Step 7: Conclusion
Once you compute this integral, you’ll have the exact length of the curve y = ln(1 + x2) from x = 0 to x = 1.6. The approximate value from numerical methods provides a good estimate if a closed-form solution isn’t attainable.