Graph the curve and visually estimate its length. Then use your calculator to find the length co

Graph the curve and visually estimate its length. Then use...

Key Concepts

Arc Length of a Curve

The arc length of a curve quantifies the distance along a curve between two points. This concept is fundamental in calculus and applies to any smooth curve, requiring integration to sum up the infinitesimal pieces of the path.

Arc Length Formula

For a function defined as y = f(x), the length of the curve from x = a to x = b is determined using the formula L = ?[a to b] ?(1 + (dy/dx)²) dx. This formula takes into account both the horizontal and vertical changes along the curve by incorporating the derivative of the function.

Differentiation

Differentiation is used to determine the rate of change or slope of a function at any given point. In the context of finding the arc length, computing the derivative of the function is essential because it directly appears in the arc length integral as the expression (dy/dx)².

Numerical Integration

Numerical integration techniques are used to approximate the value of an integral, especially when an analytic solution is difficult or impossible to obtain. Calculators and computer software often employ numerical methods to estimate the length of a curve to a specific degree of accuracy.

Graphing Functions

Graphing functions provides a visual interpretation of their behavior, allowing one to estimate properties such as arc length. Visual estimation and analysis of the graph can serve as a preliminary check or intuition behind the more precise calculations performed using the arc length formula.

Transcript

00:02 Alright, so we have to calculate, first we have to make the graph for the function which is given as y -gios to x squared plus x -cube, okay? where x is from 1 to 2.

00:20 This is from 1 to 2.

00:22 First we have to make the graph of this and we have to estimate its length visually for this values of x.

00:29 And then we have to calculate the length correct to four decimal places.

00:34 Okay? so, let's make it graph.

00:38 How we make it graph? so we have clear that idea.

00:45 First make y -axis.

00:46 Let's make x -axis.

00:47 So that is x -axis and that is y -exes.

00:52 Put x -go -1 in the graph so you'll be getting, when you put x -go -1, so you'll be getting 1 plus 1.

00:58 So the first point is, if x is equal to 1.

01:00 To 1 so i will be equal to 2 so that is our point where the starting point okay next point when y equals to 2 if you put x is equals to 2 if you put x is equals to 2 in the graph so if you put x it goes to 2 let me write down the coordinate of this point as well so the coordinate of this point is 1 comma 2 okay if i put x equals to 2 here so i'll be getting 4 plus 8 so it's essentially 12 so that will be too much that will be around here okay so this will be essentially 2 comma 2l will be the final coordinate.

01:38 Now to make this graph, how the graph will look like? if we differentiate this curve, okay? if we differentiate this curve, so what we'll be getting is, we'll be adding, y -dice will be equal to 2x plus 3x square.

01:55 So that is first derivative.

01:57 And we have to differentiate.

01:58 First derivative will tell about the graph is increasing or decreasing.

02:02 And from here we can clearly see it is actually increasing and we'll differentiate it second time as well.

02:08 So that is 2 plus 6x.

02:12 And for these values of x essentially, this is positive.

02:18 When second derivative is positive, then the graph will be actually concave upward.

02:23 Okay.

02:24 So the graph of this will be something like this...