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.
b. ∫a^b √1+36 cos² 2xdx

b. ∫a^b √1+36 cos² 2xdx

Accepted Answer

Welcome back, everyone. Find the family of differentiable functions F whose arc length on the interval from A to B inclusive is given by the integral from A to B of square root of 1 + 16 square of 5 XD X. For this problem, let's recall the R clan formula. L is equal to the integral from A to B. of square root of 1 plus. Y squared X. Now, by analogy, we can equate this equation to the given integral from A to B of square root. Of 1 + 16 sin squared of 5 XDX. We can now show that our derivative squared, specifically y squared. Is equal to 16 squared of 5 X. So what we're going to do first of all is take square roots of both sides. We got Y equals plus or minus, right, because square root of a square gives us the absolute value of that specific term, in this case Y. So we indeed have two solutions. And then we get plus or minus square root of 16 square of 5 X. Which gives us plus or minus, square root of 16 is 4, and square root of sins where there's just sign, right? We already have our signs plus or minus. So we can say plus or minus. For sine of 5 X. So these are two possible derivatives, and we want to identify the function y itself. Soy is going to be the integral of Y. We have to take the inside derivative of a derivative, meaning we want to integrate plus or minus 4 multiplied by sin of 5 XD X. And now let's go ahead and integrate. Factoring out the constant, we get plus or minus 4. The integral of sine is negative cosine. We can just include cosine of 5 X because we have two possible signs. Instead of plus or minus, we can just say minus or plus relative to our original expression, but this is not so relevant, right? We can just say plus or minus still. And now, because we have a composite function with a linear term, we also want to divide by the slope of the linear term, right? So we're dividing by 5. And we're adding a constant of integration C. Well, then, so we have identified a family of functions Y. And what we want to do is simply report them as F, right? Because this is how our problem defines them. So we can say that the family of functions F of X corresponds to + or minus 4/5. Cosine of 5 X plus a constant of integration C. That's our final answer and 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.b. ∫a^b √1+36 cos² 2xdx

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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.
b. ∫a^b √1+36 cos² 2xdx