To determine Calculate the surface area generated by revolving the curve about x axis. Explanation Given information: The curve equation is y = 2 x − x 2 . The interval is 0.5 ≤ x ≤ 1.5 . Revolution about x axis: If the function f ( x ) ≥ 0 is continuously differentiable on [ a , b ] the area of the surface generated by revolving the graph of y = f ( x ) about the x axis is S = ∫ a b 2 π y 1 + ( d y d x ) 2 d x Calculation: Show the equation as shown below: y = 2 x − x 2 = ( 2 x − x 2 ) 1 2 (1) Substitute 0.5 for x . y = 2 ( 0.5 ) − ( 0.5 ) 2 = 0.75 = 0.866 Similarly calculate the remaining values and tabulate as in Table 1. x y 0.5 0.866 0.75 0.968 1 1 1.5 0.866 Sketch the graph of the curve as shown in Figure 1. Differentiate Equation (1) with respect to y . d y d x = d d y ( ( 2 x − x 2 ) 1 2 ) = 1 2 ( 2 x − x 2 ) 1 2 − 1 ( 2 − 2 x ) = 1 2 ( 2 x − x 2 ) − 1 2 × 2 ( 1 − x ) = 1 − x 2 x − x 2 Take square on both sides

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Chapter 6.4, Problem 15E

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Calculate the surface area generated by revolving the curve about x axis.

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Chapter 6.4, Problem 14E

Chapter 6.4, Problem 16E

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