Arc Length Calculate the arc length form by the curves: 1....

Arc Length Calculate the arc length form by the curves: 1. $24xy = x^4 + 48; x \in [2,4]$ Sol. $(rac{17}{6})$ 2. $y = \frac{1}{2}a(e^{x/a} + e^{-x/a}); x \in [0, a]$ Sol. $(rac{1}{2}a(e - \frac{1}{e}))$ 3. $y = \ln x; x \in [1, 2\sqrt{2}]$ Sol. $(3 - \sqrt{2} + \ln\frac{1}{2}(2 + \sqrt{2}))$ 4. $y = \ln \cos x; x \in [\frac{\pi}{6}, \frac{\pi}{4}]$ Sol. $(\ln[(1 + \sqrt{2})/\sqrt{3}])$ 5. $x = 3y^{3/2} - 1; y \in [0, 4]$ Sol. $(\frac{8}{243}(82\sqrt{82} - 1))$

Transcript

00:01 Hi there.

00:02 We are given the curve y is equal to an x square minus 1 in this question and we need to just find the arc length.

00:11 So again for the arc length, the formula we have, which is the derivative between a to b and 1 plus the derivative of the function and take the square of this one and the x.

00:26 So first of all, let's take the derivative of the function, which is the derivative of the inside, which is 2x, divided by x square minus 1 so we have to just plug in these values into the equation and so the arc length is for this question the interval is between two to three and radical 1 plus the square of the the the square of the derivative of the function which is 2x over x square minus 1 and squared the x so we need to just calculate the integral of this equation here so first of all let's the square of that one, which is 2 to 3, square root of 1 plus 4x squared divided by, this is x squared minus 2 x squared and plus 1.

01:16 So if i just make the denominator same, so i will have radical sign, which is so x square minus 2 x squared and i mean that is sorry, that one is x to the 4th.

01:31 So let me just write again.

01:33 This is x to the 4th minus 2 x squared and plus 1 plus 4x squared divided by.

01:40 This is x squared minus 1 squared and the x.

01:48 So for the denominator of the radical sign, which is equal to, if i just simplified, we have x to the 4th plus 2 x squared plus 1 divided by x squared minus 1 squared.

02:02 We have on the denominator as radical x square one squared and divided by x square minus one squared so if i just cancel the sign and the radical um equations so we have let's say this is the x d x d x so what is left so the left uh expression is x square plus one divided by x square minus one so this is the expression that we have so we need to just find the the integral of this expression.

02:36 So what we're supposed to do, first of all, we need to just make some calculations here.

02:42 Okay, so here we can write this expression as x square minus 1 and plus 2 divided by x square minus 1, the x.

02:52 So if i just write the expression separately, which is x square minus 1 over x square minus 1 and plus 2 over x square minus 1 so so i'm not going to put the dx and you understand so these are done so what is left 1 plus 2 over so if i just factorize x square minus 1 which is x minus 1 times x plus 1 so d x and the boundaries are 2 to 3 so first of all let's take the integral of 1 which is equal to x and 2 3 and plus the integral of 2 to 3, 2 over, so i can just write the 2 on the outside of the expression.

03:35 So it is x minus 1 times x plus 1 and d x.

03:39 So let's focusing on this shaded area...

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