Un planeador viene del oeste con vientos estables. La altura del planeador arriba de la superfic

Un planeador viene del oeste con vientos estables. La altura...

Key Concepts

Integration

Integration is a key process in calculus used to accumulate quantities and compute areas under curves. When calculating the arc length of a function, integration is applied to a function derived from the original curve’s derivative, effectively summing up all the infinitesimal distances to yield the total length of the curve.

Arc Length

Arc length is a measure of the distance traveled along a curved path. The arc length of a function on a given interval is found by integrating the square root of the sum of one and the square of its derivative over that interval. This concept is fundamental in calculus for measuring the length of curves and is widely used in physics and engineering.

Derivative

The derivative of a function represents its instantaneous rate of change or the slope at any given point. In the context of arc length, the derivative is used to determine the steepness of the curve at each point, which is essential when integrating to obtain the total distance along the curve.

Substitution in Integration

Substitution is a technique used to simplify the integration process by transforming the integrand into a more manageable form. This method is particularly useful when the integrand involves composite functions or when converting complex expressions into standard forms, making the integration process more straightforward.

Transcript

00:01 Alright, so we have to calculate the length travelled by a kite, okay, and the function or the path which the kite will be followed was given as by this function, that is f of x is equals to 150 minus x minus 50 whole square divided by 40, okay? well x will be from 0 to 80 that means we have to find out the length of the path followed by the kite from x is 0 to 80 okay so for that first let's differentiate this curve so f -dice of x we'll be getting as so the differentiation of this will be actually 0 and differentiation of this will be minus 2 by 40 into x minus 50 okay to simplify it so we'll be getting as so the differentiation of this will be getting a difference 0.

00:59 1 by 20 so essentially it will be minus 1 by 20 into x minus 50 so that is the first derivative so let's calculate the length traveled by the height okay so l will be integration 0 to 80 okay root over 1 plus first derivative 1 square so first derivative is this whole square of that will be essentially x minus 50 whole square divided by 20 whole square into d of x.

01:37 Okay, so we have to integrate this function and calculate the total length travel by the kind.

01:42 So for that you have to do substitution.

01:44 So let u as equals to 1 by 20 into x minus 50.

01:59 Okay, so this is u.

02:00 So d you will become, du will become essentially differentiation of x will be 1.

02:08 So d will become d x divided by 20 okay so let's put these values so l will be essentially we'll change the limits as well so if you put the x is equals to zero the lower limit here so zero minus 50 so it will be minus 50 by 20 so the lower limit will change to minus 5 and for upper limit you have to put x is equal to 80 so put 80 minus 50 will be 30 so 30 by 20 will be 3 by 2 okay and the value here we are going to write it down as essentially this will be you assume this quantity as u so the integral will become 1 by u square and d u will be equal to d x by 20 so we can write it down as d x by okay so we have to integrate this so l will be essentially so the integration of this we can write it down as integration of okay, sorry, that's a mistake we have done actually.

03:23 So it should be actually 20 because the value of dx will be 20 d...

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