Section 9.4 Area of a surface of revolutionA surface of revolution is formed when a curve is rotated about a line.Let’s start with some simple surfaces.The lateral surface area of a circular cylinder with base radius r and height h isA = 2πrhThe lateral surface area of a circular cone with base radius r and slant height l isA = πrlThe lateral surface area of a band (frustum of a cone) with slant height l, upper radius r1andand lower radius r2isA = 2πrl, here r =r1+ r22Now we consider the surface which is obtained by rotating the curve y = f(x), a ≤ x ≤ babout the x-axis, f(x) > 0 for all x in [a, b] and f0(x) is continuous.We take a partition P of [a, b] by points a = x0< x1< ... < xn= b, and let yi= f(xi),so that the point Pi(xi, yi) lies on the curve. The part of the surface between xi−1and xiisapproximated by taking the line segment Pi−1Piand rotating it about the x-axis. The resultis a band with slant height |Pi−1Pi| and average radius r =12(yi−1+ yi), its surface area isSi= 2πyi−1+ yi2|Pi−1Pi|We know that|Pi−1Pi| =p(∆xi)2+ (∆yi)2=p1 + [f0(x∗i)]2∆xiwhere x∗i∈ [xi−1, xi]. Since ∆xiis small, we have yi= f(xi) ≈ f(x∗i) and also yi−1= f(xi−1) ≈f(x∗i) since f is continuous.Si≈ 2πf(x∗i)p1 + [f0(x∗i)]2∆xi1Thus, the area of the complete surface isSX= 2π limkP k→0nXi=1f(x∗i)p1 + [f0(x∗i)]2∆xi=2πZbaf(x)p1 + [f0(x)]2dx = 2πZbaf(x)s1 +dydx2dxIf the curve is described as x = g(y), c ≤ y ≤ d, then the formula for the surface area isSX= 2πZdcyp1 + [g0(y)]2dy = 2πZdcys1 +dxdy2dyLet’s a curve C is defined by the equationsx = x(t), y = y(t), a ≤ t ≤ bThe area of the surface generated by rotating C about x-axis isSX= 2πbZay(t)sdxdt2+dydt2dtFor rotation about the y-axis, the surface area formulas are:if the curve is given as y = f(x), a ≤ x ≤ b, then the formula for the surface area isSY= 2πbZaxs1 +dydx2dxif the curve is described as x = g(y), c ≤ y ≤ d, then the formula for the surface area isSY= 2πZdcg(y)s1 +dxdy2dyIf a curve C is defined by the equationsx = x(t), y = y(t), a ≤ t ≤ bthen the area of the surface generated by rotating C about y-axis isSY= 2πbZax(t)sdxdt2+dydt2dt2Example 1. Find the area of the surface obtained by rotating the curve about x-axis(a) y =√x, 4 ≤ x ≤ 9(b) y2= 4x + 4, 0 ≤ x ≤ 8(c) x(t) = a cos3t, y(t) = a sin3t, 0 ≤ t ≤ π/2, a is a constant.3For rotation about the y-axis, the surface area formulas are:if the curve is given as y = f(x), a ≤ x ≤ b, then the formula for the surface area isSY= 2πbZaxs1 +dfdx2dxif the curve is described as x = g(y), c ≤ y ≤ d, then the formula for the surface area isSY= 2πZdcg(y)s1 +dgdy2dyif the is defined by the equations x = x(t),y = y(t), a ≤ t ≤ b, then the area of the surface isSY=bZax(t)sdxdt2+dydt2dtExample 2. Find the area of the surface obtained by rotating the curve about y-axis(a) x =p2y −y2, 0 ≤ y ≤ 1(b) y = 1 − x2, 0 ≤ x ≤ 14(c) x = et− t, y = 4et/2, 0 ≤ t ≤
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