Find the arc length of the line y = 2x+1 on [1, 5] using calcu...

Question

Find the arc length of the line y = 2x+1 on [1, 5] using calculus and verify your answer using geometry.

Accepted Answer

Welcome back, everyone. In this problem, we want to calculate the arc length of the line Y equals X + 4 and the interval 27 using calculus and verify with the geometric distance. Now how can we find the arc length of the line using calculus? What do we know? Well, recall that the arc length formula says that the arc length L is equal to the definite integral between the bonds of A and B of the square root of 1 plus the derivative of our function squared, all with respect to X. Now, if we think about the information that we have here, if we let Y be equal to FFX, which is negative X + 4, OK. Then that means the derivative of FF X is going to be equal to -1. We also know that our lower bound A is too. And our upper bound B equals 7. So if we apply that to our formula for the arc length, that means the arc length will be the definite integral between the bones of 2 and 7 of the square root of 1 plus the derivative of F of X-1 squared with respect to X. -1 squared is positive, so this is really the definite integral of root 2 with respect to X. Now when we differentiate sorry, when we integrate route 2, OK, it gives us root to X or route 2 multiplied by X. So this is root 2 multiplied by X, OK, between 2 and 7, and we know that that will be equal to route 2 multiplied by 7 minus 2, which is equal to 5 route 2 units. Thus our arc length L is 5 route 2 units. But now remember we're told we would like to verify this with the geometric distance. What do we know about the geometric distance that can help us? Well, recall, OK, that for a geometric distance it tells us that the distanced would be equal to the square root of the difference between the X coordinates squared, the Xordinates, plus the difference between the Yordinates squared, OK? So if we think about it here. OK. When X equals 2, because you know, remember we're trying to find, we'll need the end points of the line to figure this out. And when X equals 2, Y equals -2 + 4, which equals 2. On the other hand, when X equals 7. Then Y equals -7 + 4. Which equals -3. So this tells us that 22 and 7-3 are our end points. Therefore, That means our geometric distance D will be equal to the square root of the difference between the Xordinates 7 minus 22 plus the difference between the Yordinates that's -3 minus 22. And now in this case, 7 minus 2 is 5 and 52 is 25, while -3 minus 2 is -5 and 5 squad is also 25. 25 plus 25 equals 50, and the square root of 50 can be rewritten as 5 root 2. Thus, both methods confirm that our arc length L, OK? Our arc length L is equal to 5 root 2 units. Thanks a lot for watching everyone. I hope this video helped.

Find the arc length of the line y = 2x+1 on [1, 5] using calculus and verify your answer using geometry.

Key Concepts

Arc Length Formula

Derivative of a Function

Geometric Interpretation of Line Segment Length

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