Find the volume of revolution generated by revolving the region between [tex]y=x^2[/tex] and [tex]y=2x[/tex] about the x-axis.
A. [ ] [tex]\frac{63}{15} \pi[/tex]
B. [ ] [tex]-\frac{63}{5} \pi[/tex]
C. [ ] [tex]\frac{64}{5} \pi[/tex]
D. [ ] [tex]\frac{64}{15} \pi[/tex]
Answer (1)
Find the intersection points of y = x 2 and y = 2 x , which are x = 0 and x = 2 .
Set up the volume integral using the disk method: V = π ∫ 0 2 (( 2 x ) 2 − ( x 2 ) 2 ) d x .
Evaluate the integral: V = π ∫ 0 2 ( 4 x 2 − x 4 ) d x = π [ 3 4 x 3 − 5 1 x 5 ] 0 2 .
Calculate the volume: V = 15 64 π . The volume of revolution is 15 64 π .
Explanation
Problem Analysis We are asked to find the volume of the solid generated by revolving the region between the curves y = x 2 and y = 2 x about the x-axis.
Finding Intersection Points First, we need to find the points of intersection of the two curves. We set x 2 = 2 x , which gives us x 2 − 2 x = 0 . Factoring, we get x ( x − 2 ) = 0 , so the points of intersection occur at x = 0 and x = 2 . These will be our limits of integration.
Setting up the Integral Next, we use the disk method to find the volume of revolution. The volume is given by the integral V = π ∫ a b ( R ( x ) 2 − r ( x ) 2 ) d x , where R ( x ) is the outer radius and r ( x ) is the inner radius. In this case, the outer radius is R ( x ) = 2 x and the inner radius is r ( x ) = x 2 .
Defining the Volume Integral Now we set up the integral with the limits of integration and the radii: V = π ∫ 0 2 (( 2 x ) 2 − ( x 2 ) 2 ) d x = π ∫ 0 2 ( 4 x 2 − x 4 ) d x
Evaluating the Integral Now we evaluate the integral: V = π [ 3 4 x 3 − 5 1 x 5 ] 0 2 = π ( 3 4 ( 2 ) 3 − 5 1 ( 2 ) 5 ) − π ( 3 4 ( 0 ) 3 − 5 1 ( 0 ) 5 ) V = π ( 3 4 ( 8 ) − 5 1 ( 32 ) ) = π ( 3 32 − 5 32 ) = π ( 15 160 − 96 ) = π ( 15 64 ) V = 15 64 π
Final Volume Therefore, the volume of the solid of revolution is 15 64 π .
Examples
Imagine you are designing a funnel. The curves y = x 2 and y = 2 x define the shape of the funnel's cross-section. By revolving this region around the x-axis, you create a 3D funnel. Calculating the volume of this funnel using the method of disks helps you determine how much liquid it can hold, which is crucial for practical applications. This method is also used in various engineering applications, such as designing tanks and containers with specific volume requirements.
