Find the volume of the solid of revolution formed by rotating the region bounded by [tex]y=x^3[/tex] and [tex]y=x[/tex] about the line [tex]x=-2[/tex]. Use the shell method. V = [?]
Round your answer to the nearest thousandth.
Answer (2)
Find the intersection points of y = x 3 and y = x , which are x = − 1 , 0 , 1 .
Set up the volume integral using the shell method: V = 4 π ∫ 0 1 ( x + 2 ) ( x − x 3 ) d x .
Evaluate the integral: ∫ 0 1 ( x + 2 ) ( x − x 3 ) d x = 30 19 .
Calculate the volume: V = 4 π ( 30 19 ) = 15 38 π ≈ 7.959 .
Explanation
Problem Setup We are asked to find the volume of the solid of revolution formed by rotating the region bounded by the curves y = x 3 and y = x about the line x = − 2 . We will use the shell method to find the volume.
Finding Intersection Points First, we need to find the intersection points of the curves y = x 3 and y = x . We set x 3 = x , which gives x 3 − x = 0 . Factoring, we get x ( x 2 − 1 ) = x ( x − 1 ) ( x + 1 ) = 0 . Thus, the intersection points occur at x = − 1 , 0 , 1 .
Setting up the Shell Method The region is bounded by y = x 3 and y = x between x = − 1 and x = 1 . Since we are rotating about the line x = − 2 , the radius of a cylindrical shell is x − ( − 2 ) = x + 2 . The height of the shell is given by the difference between the two curves, ∣ x − x 3 ∣ .
Setting up the Integral The volume of the solid of revolution is given by the integral V = 2 π ∫ − 1 1 ( x + 2 ) ∣ x − x 3 ∣ d x . Since the region is symmetric with respect to the origin, we can integrate from 0 to 1 and multiply by 2. Thus, V = 4 π ∫ 0 1 ( x + 2 ) ( x − x 3 ) d x .
Expanding the Integrand Now, we evaluate the integral: ∫ 0 1 ( x + 2 ) ( x − x 3 ) d x = ∫ 0 1 ( x 2 − x 4 + 2 x − 2 x 3 ) d x .
Evaluating the Integral We compute the definite integral: ∫ 0 1 ( x 2 − x 4 + 2 x − 2 x 3 ) d x = [ 3 x 3 − 5 x 5 + x 2 − 2 x 4 ] 0 1 = 3 1 − 5 1 + 1 − 2 1 = 30 10 − 6 + 30 − 15 = 30 19 .
Calculating the Volume Multiply by 4 π to find the volume: V = 4 π ( 30 19 ) = 15 38 π .
Approximating the Volume Finally, we approximate the volume to the nearest thousandth: V ≈ 15 38 × 3.14159 ≈ 7.9587 . Rounding to the nearest thousandth, we get V ≈ 7.959 .
Final Answer The volume of the solid of revolution is approximately 7.959 .
Examples
Imagine you're designing a custom-shaped fuel tank for a rocket. The tank's shape is formed by rotating a region bounded by specific curves around an axis. By calculating the volume of this solid of revolution, you can determine exactly how much fuel the tank will hold. This is crucial for mission planning and ensuring the rocket has enough fuel to reach its destination. Understanding the shell method and volume calculations allows for precise design and efficient use of space in engineering applications.
Using the shell method, we calculated the volume by integrating the height and radius of cylindrical shells formed by rotating the area between the curves y = x 3 and y = x around the line x = − 2 . The final volume of the solid of revolution is approximately 7.959 .
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