Chapter 6 | Applications of Integration
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6.20 Let g ( y ) =1/ y . Calculate the arc length of the graph of g ( y ) over the interval [1, 4]. Use a computer or calculator to approximate the value of the integral.
Area of a Surface of Revolution The concepts we used to find the arc length of a curve can be extended to find the surface area of a surface of revolution. Surface area is the total area of the outer layer of an object. For objects such as cubes or bricks, the surface area of the object is the sum of the areas of all of its faces. For curved surfaces, the situation is a little more complex. Let f ( x ) be a nonnegative smooth function over the interval ⎡ ⎣ a , b ⎤ ⎦ . We wish to find the surface area of the surface of revolution created by revolving the graph of y = f ( x ) around the x -axis as shown in the following figure.
Figure 6.40 (a) A curve representing the function f ( x ). (b) The surface of revolution formed by revolving the graph of f ( x ) around the x -axis.
As we have done many times before, we are going to partition the interval ⎡ ⎣ a , b ⎤ ⎦ and approximate the surface area by calculating the surface area of simpler shapes. We start by using line segments to approximate the curve, as we did earlier in this section. For i =0, 1, 2,…, n , let P ={ x i } be a regular partition of ⎡ ⎣ a , b ⎤ ⎦ . Then, for i =1, 2,…, n , construct a line segment from the point ⎛ ⎝ x i −1 , f ( x i −1 ) ⎞ ⎠ to the point ⎛ ⎝ x i , f ( x i ) ⎞ ⎠ . Now, revolve these line segments around the x -axis to generate an approximation of the surface of revolution as shown in the following figure.
Figure 6.41 (a) Approximating f ( x ) with line segments. (b) The surface of revolution formed by revolving the line segments around the x -axis.
Notice that when each line segment is revolved around the axis, it produces a band. These bands are actually pieces of cones
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