Video Transcript
Write, but do not evaluate, an integral for the arc length of the curve 𝑦 is equal to the natural logarithm of two 𝑥 to the fifth power plus seven between 𝑥 is equal to one and 𝑥 is equal to four.
The question gives us a curve and it wants us to write an integral which gives us the arc length of this curve between the values of 𝑥 is equal to one and 𝑥 is equal to four. And we know how to find the arc length of certain curves. We know if 𝑓 prime is a continuous function on the closed interval from 𝑎 to 𝑏. Then the length of the curve 𝑦 is equal to 𝑓 of 𝑥 between the values of 𝑥 is equal to 𝑎 and 𝑥 is equal to 𝑏 is equal to the integral from 𝑎 to 𝑏 of the square root of one plus 𝑓 prime of 𝑥 squared with respect to 𝑥.
We want the arc length of the curve 𝑦 is equal to the natural logarithm of two 𝑥 to the fifth power plus seven between the values of 𝑥 is equal to one and 𝑥 is equal to four. So we’ll set our function 𝑓 of 𝑥 to be the natural logarithm of two 𝑥 to the fifth power plus seven. We’ll set 𝑎 equal to one and 𝑏 equal to four. Then, to use this arc length formula, we just need to show that 𝑓 prime is continuous on the closed interval from 𝑎 to 𝑏. So let’s find an expression for 𝑓 prime of 𝑥.
We can see if we try to differentiate 𝑓 of 𝑥, we need to differentiate a composite function. So we’ll do this by using the chain rule. If we set 𝑢 of 𝑥 to be our inner function two 𝑥 to the fifth power, then we can see that 𝑓 is a function of 𝑢 and 𝑢 is a function of 𝑥. Then by the chain rule, the derivative of 𝑓 of 𝑢 of 𝑥 is equal to 𝑓 prime evaluated at 𝑢 of 𝑥 times 𝑢 prime of 𝑥. Therefore, using the chain rule, we have 𝑓 prime of 𝑥 is equal to the derivative of the natural logarithm of 𝑢 with respect to 𝑢 times the derivative of two 𝑥 to the fifth power with respect to 𝑥.
And we can evaluate each of these derivatives separately. The derivative of the natural logarithm of 𝑢 with respect to 𝑢 is one over 𝑢. And the derivative of two 𝑥 to the fifth power with respect to 𝑥 is 10𝑥 to the fourth power. Remember, this is an expression for 𝑓 prime of 𝑥. So we should give our answer in terms of 𝑥. We’ll do this by using our substitution 𝑢 is equal to two 𝑥 to the fifth power. Using this, we get 10𝑥 to the fourth power divided by two 𝑥 to the fifth power, which simplifies to give us five over 𝑥. So we found that 𝑓 prime of 𝑥 is equal to five over 𝑥. This is a rational function. And we know that all rational functions are continuous on their domain. In fact, we can see the domain of 𝑓 prime of 𝑥 is all values of 𝑥 not equal to zero.
Remember, we wanted 𝑓 prime to be continuous on our closed interval from one to four. This doesn’t contain 𝑥 is equal to zero. So our function 𝑓 prime is continuous on this interval. Therefore, we’ve justified the use of our integral formula to find the length of our arc. Substituting in 𝑓 prime of 𝑥 as five over 𝑥, 𝑎 is equal to one, and 𝑏 is equal to four, we get the length of our arc is equal to the integral from one to four of the square root of one plus five over 𝑥 squared with respect to 𝑥. And the last thing we’ll do is simplify five over 𝑥 all squared to give us 25 over 𝑥 squared.
Therefore, we’ve shown we can find the arc length of the curve 𝑦 is equal to the natural logarithm of two 𝑥 to the fifth power plus seven between the values of 𝑥 is equal to one and 𝑥 is equal to four. By evaluating the integral from one to four of the square root of one plus 25 over 𝑥 squared with respect to 𝑥.
