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Write an integral expression that will give the length of the path given by from to .
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Q. Write an integral expression that will give the length of the path given by from to .
- Set Up Integral for Arc Length: To find the length of the path described by the function from to , we need to set up the integral for arc length in one dimension. The formula for the arc length of a function from to is given by the integral from to of the square root of plus the derivative of squared, .
- Find Derivative of Function: First, we need to find the derivative of the function . The derivative, , is found using the power rule for differentiation.
- Square the Derivative: Next, we square the derivative to include it in the arc length formula.(f'(x))^\(2 = (x^)^ = x^\.
- Write Integral Expression: Now, we can write the integral expression for the arc length. The integral expression is the integral from to of the square root of dx.So, the integral expression is .
- Evaluate Integral for Length: This integral expression represents the length of the path of the function from to . To find the actual length, one would need to evaluate this integral, which may require numerical methods or approximation techniques as it does not have a simple antiderivative.
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