JUST THE MATHS UNIT NUMBER INTEGRATION APPLICATIONS 6 (First moments of an arc) A.J.Hobson

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1 JUST THE MATHS UNIT NUMBER 13.6 INTEGRATION APPLICATIONS 6 (First moments of n rc) by A.J.Hobson Introduction First moment of n rc bout the y-xis First moment of n rc bout the x-xis The centroid of n rc Exercises Answers to exercises

2 UNIT INTEGRATION APPLICATIONS 6 FIRST MOMENTS OF AN ARC INTRODUCTION Suppose tht C denotes n rc (with length s) in the xy-plne of crtesin co-ordintes; nd suppose tht δs is the length of smll element of this rc. Then the first moment of C bout fixed line, l, in the plne of C is given by lim δs C hδs, where h is the perpendiculr distnce, from l, of the element with length δs. h l δs C FIRST MOMENT OF AN ARC ABOUT THE Y-AXIS Let us consider n rc of the curve, whose eqution is y = f(x), joining two points, P nd Q, t x = nd x = b, respectively. 1

3 y δy P δs Q O δx x b The rc my divided up into smll elements of typicl length, δs, by using neighbouring points long the rc, seprted by typicl distnces of δx (prllel to the x-xis) nd δy (prllel to the y-xis). The first moment of ech element bout the y-xis is x times the length of the element; tht is xδs, implying tht the totl first moment of the rc bout the y-xis is given by lim δs C xδs. But, from Pythgors Theorem, δs ( ) (δx) + (δy) δy = 1 + δx, δx so tht the first moment of the rc becomes lim δx x=b x= ( ) δy x 1 + δx = δx b ( ) x 1 + dx. dx Note: If the curve is given prmetriclly by x = x(t), y = y(t),

4 then, using the sme principles s in Unit 13.4, we my conclude tht the first moment of the rc bout the y-xis is given by t ± t 1 ) ( dx x + ( ), ccording s dx is positive or negtive FIRST MOMENT OF AN ARC ABOUT THE X-AXIS () For n rc whose eqution is y = f(x), contined between x = nd x = b, the first moment bout the x-xis will be b ( ) y 1 + dx. dx Note: If the curve is given prmetriclly by x = x(t), y = y(t), then, using the sme principles s in Unit 13.4, the first moment of the rc bout the x-xis is given by t ± t 1 ) ( dx y + ( ), ccording s dx is positive or negtive. 3

5 (b) For n rc whose eqution is x = g(y), contined between y = c nd y = d, we my reverse the roles of x nd y in section so tht the first moment of the rc bout the x-xis is given by d c ( ) dx y 1 +. d y S δy δs c R O δx x Note: If the curve is given prmetriclly by x = x(t), y = y(t), then, using the sme principles s in Unit 13.4, we my conclude tht the first moment of the rc bout the x-xis is given by t ± t 1 ) ( dx y + ( ), ccording s is positive or negtive nd where t = t 1 when y = c nd t = t when y = d. 4

6 EXAMPLES 1. Determine the first moments bout the x-xis nd the y-xis of the rc of the circle, with eqution x + y =, lying in the first qudrnt. Solution y O x Using implicit differentition, we hve x + y dx =, nd hence, dx = x y. The first moment of the rc bout the y-xis is therefore given by x But x + y = nd y = x. Hence, 1 + x y dx = x x y + y dx. first moment = x ] [ x dx = ( x ) =. By symmetry, the first moment of the rc bout the x-xis will lso be. 5

7 . Determine the first moments bout the x-xis nd the y-xis of the first qudrnt rc of the curve with prmetric equtions Solution x = cos 3 θ, y = sin 3 θ. y O x Firstly, we hve dx dθ = 3cos θ sin θ nd dθ = 3sin θ cos θ. Hence, the first moment bout the x-xis is given by π which, on using cos θ + sin θ 1, becomes y 9 cos 4 θsin θ + 9 sin 4 θcos θ dθ, π sin 3 θ.3 cos θ sin θ dθ = 3 π sin 4 θ cos θ dθ [ sin = 3 5 ] π θ 5 =

8 Similrly, the first moment of the rc bout the y-xis is given by π ) ( ) ( dx π x + dθ = cos 3 θ.(3 cos θ sin θ) dθ dθ dθ = 3 π cos 4 θ sin θ dθ = 3 [ ] π cos5 θ 5 = 3 5, though, gin, this second result could be deduced, by symmetry, from the first THE CENTROID OF AN ARC Hving clculted the first moments of n rc bout both the x-xis nd the y-xis it is possible to determine point, (x, y), in the xy-plne with the property tht () The first moment bout the y-xis is given by sx, where s is the totl length of the rc; nd (b) The first moment bout the x-xis is given by sy, where s is the totl length of the rc. The point is clled the centroid or the geometric centre of the rc nd, for n rc of the curve with eqution y = f(x), between x = nd x = b, its co-ordintes re given by x = b 1 x + ( ) dx dx b 1 + ( ) nd y = dx dx b 1 y + ( ) dx dx b 1 + ( ). dx dx Notes: (i) The first moment of n rc bout n xis through its centroid will, by definition, be zero. In prticulr, if we tke the y-xis to be prllel to the given xis, with x s the perpendiculr distnce from n element, δs, to the y-xis, the first moment bout the given xis will be C (x x)δs = C xδs x C δs = sx sx =. (ii) The centroid effectively tries to concentrte the whole rc t single point for the purposes of considering first moments. In prctice, it corresponds, for exmple, to the position of the centre of mss of thin wire with uniform density. 7

9 EXAMPLES 1. Determine the crtesin co-ordintes of the centroid of the rc of the circle, with eqution x + y =, lying in the first qudrnt. Solution y O x From n erlier exmple in this unit, we know tht the first moments of the rc bout the x-xis nd the y-xis re both equl to. Also, the length of the rc is π, which implies tht x = π nd y = π.. Determine the crtesin co-ordintes of the centroid of the first qudrnt rc of the curve with prmetric equtions x = cos 3 θ, y = sin 3 θ. 8

10 Solution y O x From n erlier exmple in this unit, we know tht dx dθ = 3cos θ sin θ nd dθ = 3sin θ cos θ nd tht the first moments of the rc bout the x-xis nd the y-xis re both equl to 3 5. Also, the length of the rc is given by π This simplifies to ) ( dx + dθ ( ) dθ = dθ π 9 cos 4 θsin θ + 9 sin 4 θcos θ dθ. π 3 cos θ sin θ dθ = 3 [ sin θ ] π = 3. Thus, x = 5 nd y = 5. 9

11 EXERCISES 1. Determine the first moment bout the y-xis of the rc of the curve with eqution lying between x = nd x = 1. y = x,. Determine the first moment bout the x-xis of the rc of the curve with eqution lying between y =.1 nd y =.5. x = 5y, 3. Determine the first moment bout the x-xis of the rc of the curve with eqution lying between x = 3 nd x = 4. y = x, 4. Verify, using integrtion, tht the centroid of the stright line segment, defined by the eqution y = 3x +, from x = to x = 1, lies t its centre point. 5. Determine the crtesin co-ordintes of the centroid of the rc of the circle given prmetriclly by from θ = π 6 to θ = π For the curve whose eqution is x = 5 cos θ, y = 5 sin θ, 9y = x(3 x), show tht dx = 1 x x. Hence ( show tht the centroid of the first qudrnt rch of this curve lies t the point 7, )

12 ANSWERS TO EXERCISES x = 1 nd y = x = 15 π 4.77, y =. 11