Find the surface area obtained by revolving the curve [tex]$y=x^3+1$[/tex] between [tex]$(-1,0)$[/tex] and [tex]$(1,2)$[/tex] about the [tex]$x$[/tex]-axis.
Round your answer to the nearest thousandth.
Answer (2)
Set up the surface area integral: A = 2 π ∫ − 1 1 ( x 3 + 1 ) 1 + ( 3 x 2 ) 2 d x .
Simplify the integral: A = 2 π ∫ − 1 1 ( x 3 + 1 ) 1 + 9 x 4 d x .
Evaluate the definite integral numerically: ∫ − 1 1 ( x 3 + 1 ) 1 + 9 x 4 d x ≈ 3.096 .
Calculate the surface area: A = 2 π × 3.096 ≈ 19.451 .
Explanation
Problem Setup We are asked to find the surface area obtained by revolving the curve y = x 3 + 1 between x = − 1 and x = 1 about the x -axis. The formula for the surface area of revolution about the x -axis is given by A = 2 π ∫ a b y 1 + ( d x d y ) 2 d x
Finding the Derivative First, we need to find the derivative of y with respect to x :
y = x 3 + 1 d x d y = 3 x 2
Substituting into the Formula Now, we substitute y = x 3 + 1 and d x d y = 3 x 2 into the surface area formula with a = − 1 and b = 1 :
A = 2 π ∫ − 1 1 ( x 3 + 1 ) 1 + ( 3 x 2 ) 2 d x = 2 π ∫ − 1 1 ( x 3 + 1 ) 1 + 9 x 4 d x
Evaluating the Integral The integral ∫ − 1 1 ( x 3 + 1 ) 1 + 9 x 4 d x is difficult to evaluate analytically. Therefore, we will use numerical integration to approximate the value of the definite integral. Using a numerical integration method, we find that ∫ − 1 1 ( x 3 + 1 ) 1 + 9 x 4 d x ≈ 3.096
Calculating the Surface Area Finally, we multiply the result by 2 π to get the surface area: A = 2 π × 3.096 ≈ 19.451
Final Answer Therefore, the surface area obtained by revolving the curve y = x 3 + 1 between x = − 1 and x = 1 about the x -axis, rounded to the nearest thousandth, is approximately 19.451 .
Examples
Imagine you are designing a vase with a curved profile described by the function y = x 3 + 1 . To estimate the amount of material needed to create the vase, you need to calculate the surface area. By revolving this curve around the x-axis, you create the vase's shape. The surface area calculation, as performed in this problem, allows you to determine the amount of glass or ceramic required, helping to optimize material usage and reduce costs. This is a practical application of calculus in manufacturing and design.
The surface area generated by revolving the curve y = x 3 + 1 about the x -axis from − 1 to 1 is calculated using the surface area formula for revolution. This results in a surface area of approximately 19.451 square units. This calculation involves evaluating a definite integral numerically due to its complexity.
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