Revolving the curve [tex]y=-x^2+1[/tex] between [tex](0,1)[/tex] and [tex](1,0)[/tex] about the [tex]x[/tex]-axis, [tex]SA=[?][/tex]
Round your answer to the nearest thousandth.
Answer (2)
Find the derivative of y = − x 2 + 1 , which is d x d y = − 2 x .
Substitute y and d x d y into the surface area formula: S A = 2 π ∫ 0 1 ( 1 − x 2 ) 1 + 4 x 2 d x .
Evaluate the integral numerically: ∫ 0 1 ( 1 − x 2 ) 1 + 4 x 2 d x ≈ 0.872 .
Multiply by 2 π to get the surface area and round to the nearest thousandth: S A ≈ 5.483 .
Explanation
Problem Setup We are asked to find the surface area of the solid generated by revolving the curve y = − x 2 + 1 between x = 0 and x = 1 about the x-axis.
Surface Area Formula The formula for the surface area of revolution about the x-axis is given by: S A = 2 π ∫ a b y 1 + ( d x d y ) 2 d x where y = f ( x ) is the curve being revolved, and a and b are the limits of integration.
Finding the Derivative In our case, y = − x 2 + 1 , so we need to find the derivative d x d y :
d x d y = − 2 x
Substituting into the Formula Now, we substitute y and d x d y into the surface area formula. The limits of integration are a = 0 and b = 1 :
S A = 2 π ∫ 0 1 ( − x 2 + 1 ) 1 + ( − 2 x ) 2 d x = 2 π ∫ 0 1 ( 1 − x 2 ) 1 + 4 x 2 d x
Evaluating the Integral The integral ∫ 0 1 ( 1 − x 2 ) 1 + 4 x 2 d x is difficult to evaluate analytically, so we will use numerical methods to approximate it. Using a numerical integration method (Simpson's rule), we find that: ∫ 0 1 ( 1 − x 2 ) 1 + 4 x 2 d x ≈ 0.872
Calculating the Surface Area Now, we multiply the result of the integral by 2 π to get the surface area: S A = 2 π × 0.872 ≈ 5.483
Final Answer Rounding the final answer to the nearest thousandth, we get: S A ≈ 5.483
Examples
Surface area calculations are crucial in various fields. For instance, in architecture, determining the surface area of curved structures is essential for estimating material costs like paint or cladding. Similarly, in chemical engineering, the surface area of catalyst particles affects reaction rates, making its calculation vital for optimizing processes. This problem demonstrates how calculus can be applied to solve real-world engineering and design challenges.
The surface area of the solid formed by revolving the curve y = − x 2 + 1 about the x-axis from x = 0 to x = 1 is calculated as approximately 5.486 . This involves integrating the function using the surface area formula for revolution. The final result is rounded to the nearest thousandth.
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