Functions from arc length What differentiable functions have a...

Question

Functions from arc length What differentiable functions have an arc length on the interval [a, b] given by the following integrals? Note that the answers are not unique. Give a family of functions that satisfy the conditions.
a. ∫a^b √1+16x⁴ dx

a. ∫a^b √1+16x⁴ dx

Accepted Answer

Welcome back, everyone. Find the family of differentiable functions F whose are planned on. The interval from A to B inclusive is given by the integral from A to B of square root of 1 + 25 X raise to the power of 4DX. For this problem, let's recall the R plan formula, which is the integral from A to B of square root of 1 plus F of X squared X. What we want to do is simply equate it to the given integral from A to B, of square root of 1 + 25 X raises the power of 4DX. What we can notice is that by analogy 25 x raises the power of 4 is the derivative squared, so we can show that F of X squared must be equal to 25 x raises the power of 4. We are going to solve for F of X to begin with, which is square root of. 25 X raises the power of 4, and because we're taking square root of a 2, we are going to have two solutions, positive and negative. So now let's go ahead and take square roots, square root of 25 is 5. Square root of X raises the power of 4 is X2. So we get plus or minus 5 X squad. That's our derivative. And finally, if we want to identify the family of functions F of X. This means integrating the derivative because antiderivative of a derivativescerns the original function itself. So let's go ahead and integrate plus or minus 5 X square DX. Factoring out the constant, we get plus or minus 5, and the integral of X squared as x cubed divided by 3. We're going to add a constant of integration, see because this is an indefinite integral, right? Therefore, we can conclude that. The family of functions F corresponds to plus or minus 5/3 X cubed plus C. Thank you for watching.

Functions from arc length What differentiable functions have an arc length on the interval [a, b] given by the following integrals? Note that the answers are not unique. Give a family of functions that satisfy the conditions.
a. ∫a^b √1+16x⁴ dx

Key Concepts

Arc Length Formula

Relationship Between the Integrand and the Derivative

Family of Functions and Integration

Watch next

Functions from arc length What differentiable functions have an arc length on the interval [a, b] given by the following integrals? Note that the answers are not unique. Give a family of functions that satisfy the conditions.a. ∫a^b √1+16x⁴ dx

Key Concepts

Arc Length Formula

Relationship Between the Integrand and the Derivative

Family of Functions and Integration

Watch next

Functions from arc length What differentiable functions have an arc length on the interval [a, b] given by the following integrals? Note that the answers are not unique. Give a family of functions that satisfy the conditions.
a. ∫a^b √1+16x⁴ dx