Problem

Use Simpson's Rule with $n=10$ to estimate the ar…

09:50

Problem 20 Hard Difficulty

Graph the curve and visually estimate its length.
Then use your calculator to find the length correct to four decimal places.
$$y=x+\cos x, \quad 0 \leqslant x \leqslant \pi / 2$$

Answer

$L=\int_{0}^{\pi / 2} \sqrt{1+(1-\sin x)^{2}} d x \approx 1.7294$

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Calculus 2 / BC

Essential Calculus Early Transcendentals

Chapter 7

APPLICATIONS OF INTEGRATION

Section 4

Arc Length

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Video Transcript

Okay, so in this question we have to first calculate so if first we have to graph the curve and the curve which we have got is y is equals to x, plus cos of x, where x is from 0 to pi by 2 point. So what we have to do in this question? First, we have to graph it and then we have to approximately approximately get the length of the curve using the graft, and then we have to calculate the value of the length of the curve using our formula. So that is what we have to do. First, let's differentiate this curve, then you go to graph. The first derivative for this curve is 1 minus sine of x. I i the derivative as well, because that will also be required, or so that will be. Minus differentiation of n minus sine x will be torn to 0, and it will be in on will be cos x, okay and if you see 40 to pi by 2. This first derivative is actually positive: okay, because if in 0 to pi by 2, this sine x will be always smaller than 1 and in 0 to pi by 2, this cos x will be positive, but here we have minus cos x. So essentially, this will be negative, so first derivative is positive, but second derivative is negative. First, relative positive in this graph and second is written normal negative means concave downward okay, so, let's make it that is x when x is goes to 0 to put goes to 0 here, we'll be getting 0 here, so this will be 0. So, when x is equal to 0, this is point and coordinate of this is 0 comma 1. Next, when x, is equals to 1. If you put x equal to pi by 2 pi by 2, so we'll be getting 5 by 2 hear and we'll be getting 1 or 0 here, so the next point will be somewhere here. Coordinator of this point will be sorry pi by 2, comma, okay- and you see here, the second derivative is negative and first dive piso second dainties. It will be concave downwards, so the curve will do something like this, so that will be the graph for this. Okay. Now, let's calculate the length of the arc, so approximate length using distance formulas for length of the arc is in distance formula, so that will be root or pi by 2 minus 0. So that will be i by 2 minus 0 square plus 2 minus 1, who, if he simply if you try to calculate the value of years, so you see here pi by 2 whole square, so pi by 2 is 1.57, okay and 1 square of that, and Here, pi by 2, minus 1 point so pi by 2, minus 1 will be pi by 2. Is 1.57 minus consistency will be getting 0 point 5.5, so something whole square. So this value is 1 point fraction whole square, and this value will be 1.0 .5 whole square, so the approximate have to take under root as well. So the approximate value here we are going to wit as may be, are 1.5 using distance formula. So that is the approximate will we are going to be? Okay now, let's calculate the length of the curve using our formula length of using formula, so that will be integration 0 to pi by 2 root over 1 plus first rate whole square, and you can thence. First, root is 1 minus sine x, 1 minus sine x whole square into dx, and the approximate value of this we are writing is actually a that is approximate value. We are going to get for this. Integral is 1 point 72941.7294, and that is almost. That is approximate value, so there they are approximate. This is our estimation, and this is our approximate value. So that is the answer.

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