m23 Area of Surface of Revolution

m23 Area of Surface of Revolution - MATH023 Surface Area...

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MATH023 Surface Area
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Objectives At the end of the period, you should be able to: Compute the areas of surfaces of revolution by integration. Derive common formulas of surface areas and lateral areas of geometric figures.
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Surface Area A surface of revolution is that surface obtained by revolving an arc about a given axis of rotation
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Surface Area For the purposes of the derivation of the formula, let’s look at rotating the continuous function y=f(x) in the interval [a,b] about the x-axis. We can derive a formula for the surface area much as we derived the formula for arc length .
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Surface Area We’ll start by dividing the integral into n equal subintervals of width ∆x . On each subinterval we will approximate the function with a straight line that agrees with the function at the endpoints of the each interval.
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Surface Area Rotate the approximations about the x-axis and we get the following solid. The approximation on each interval gives a distinct portion of the solid and to make this clear each portion is colored differently. Each of these portions are called frustums and we know how to find the surface area of frustums.
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Surface Area The surface area of a frustum is given by where and l is the length of the slant of the frustum. The surface area of the frustum on the interval is approximately
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Surface Area The surface area of the whole solid is then approximately and we can get the exact surface area by taking the limit as n goes to infinity
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Surface Area When the equation of the curve is given in non- parametric form y=f(x), a
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m23 Area of Surface of Revolution - MATH023 Surface Area...

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