How can we determine the exact length of the curve defined by the equation y = ln(1 + x^2) over the interval from x = 0 to x = 1?

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To find the exact length of the curve defined by the equation y = ln(1 + x^2) over the interval from x = 0 to x = 1, we can use the formula for the length of a curve in Cartesian coordinates. The formula is given by:

L = ∫a √1 + (dy/dx)2 dx

In this case, a = 0 and b = 1. First, we need to calculate the derivative dy/dx.

Starting with the function:

y = ln(1 + x^2)

We differentiate this function:

dy/dx = &frac{d}{dx}(ln(1 + x^2)) = &frac{1}{1 + x^2} * &frac{d}{dx}(1 + x^2) = &frac{1}{1 + x^2} * 2x = &frac{2x}{1 + x^2}

Now we need to calculate (dy/dx)2:

(dy/dx)2 = &left(&frac{2x}{1 + x^2}&right)2 = &frac{4x^2}{(1 + x^2)2}

Next, we substitute this into the length formula:

L = ∫01 √1 + &frac{4x^2}{(1 + x^2)2} dx

To simplify this, we need to find a common denominator:

L = ∫01 √2 + 4x^2}{(1 + x^2)2} dx

By expanding (1 + x^2)2, we get:

(1 + x^2)2 + 4x^2 = 1 + 2x^2 + x^4 + 4x^2 = 1 + 6x^2 + x^4

Thus, the integrand becomes:

√2}} = &frac{√(1 + 6x^2 + x^4)}{1 + x^2}

Therefore, the curve length expression is:

L = ∫01 &frac{√(1 + 6x^2 + x^4)}{1 + x^2} dx

To solve this integral, we can use numerical methods or consult integral tables as it can be complex. However, once evaluated, it yields the exact length of the curve from x = 0 to x = 1.

In conclusion, finding the exact length of the curve involves derivative calculations and working through a definite integral to yield the length between the specified bounds.

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