The length of the are of the curve y= log sec x from x=0 te x=π/ 4 is (a) log, 2 (b) logy 3 (c)

The length of the are of the curve y= log sec x from x=0 te...

Key Concepts

Logarithmic Functions

Logarithmic functions have unique properties that affect both their differentiation and integration. Understanding the behavior of logarithms, especially when combined with other functions like trigonometric ones, is important for correctly applying the chain rule in differentiation. Their properties can also simplify the integrals encountered when computing the arc length, particularly when logarithmic expressions are present in the integrand.

Arc Length Formula

The arc length formula is a fundamental concept used to compute the length of a curve described by a function over a given interval. It is given by the integral L = ?[a,b] ?(1 + (dy/dx)²) dx, where dy/dx is the derivative of the function. This formula converts the differential elements of the curve into a complete length measurement, making it a key concept in differential calculus and vector calculus.

Differentiation of Functions

Differentiating functions, especially those involving logarithmic and trigonometric components, is crucial when applying the arc length formula. Using the chain rule, the derivative of composite functions like y = log(sec x) is obtained by computing the derivative of the outer function and then multiplying by the derivative of the inner function. This process is vital for correctly determining the slope of the curve at any point along the interval of integration.

Trigonometric Identities

Trigonometric identities are essential tools in simplifying expressions during differentiation and integration. In the context of the problem, identities such as tan²x + 1 = sec²x help simplify the derivative obtained from a logarithmic function involving the secant. This simplification reduces the complexity of the integrand and makes the subsequent integration process more tractable.

Transcript

00:01 Hi, we're given here this graph here this draft says the length of the r we have a parabola at a it is at a.

00:07 0 the point b is at b comma 0.

00:09 Y equal to effect.

00:10 The length of the art from a to b is given as a to b square square 2.

00:16 In parametric form, if we take x equals ft, y to of t t t t t t, if we have x equals a, then we have t equals t2.

00:25 If we have x equals t2.

00:28 The length is given as t2.

00:29 Square root at test t holsquare plus d d ddh t holes.

00:33 So that is in a metric form.

00:34 Now the first question is given as based on this.

00:38 The first question says that we need to find a length of the arc of the parabola from x equal to 0 to x equal to 1.

00:50 So if we take here from 0 to 1, let's say here we have this is 1 .0 and this is 0 .0.

00:56 So this length we need to find out here from here this length.

01:01 Let's work on this length.

01:05 So first we'll have y square equal to 2.

01:09 We have 4x or 2y, d .y over dx.

01:14 That equals 4.

01:15 Since we have got now, from here we have d, y, or dx is equal to 4 over 2y.

01:27 And here we have 2.

01:29 So from here we have d, d, y, over d x, is coming out to be.

01:32 Of 2 we have already taken that, so we'll just ignore this 2 here.

01:40 So it's going to add to d, dx, equals we have 2 over 1, which we have got now.

01:45 Now next we find out square root 1 plus d, d, d ,x, square square root we have 1 plus 4 over y square that's given as square root 1 plus 4 over we have y square square x so jvm now 4x that's coming out to be square root x plus 1 over x this is good now next we'll find out the arc length from 0 to 1 to arc length is given as 0 to 1 we have square root x plus 1 over x d d x now let's simplify this to put for that what we can do, we can reduce the number of substitution...

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