y=x^3
y=8
x=0
about x-axis
its using shell method
i need step by step help seriously !!!!
Sure! I'd be happy to help you step by step.
To use the shell method to find the volume of the solid generated when the region between the curves y = x^3, y = 8, and the line x = 0 is revolved around the x-axis, we need to follow these steps:
Step 1: Identify the limits of integration.
- The limits of integration will determine the range of x-values that we need to consider. In this case, since the solid is revolved around the x-axis, we need to find the x-values at which the curves intersect.
- The equation y = 8 represents a horizontal line, so it intersects the curve y = x^3 at just one point. To find this point, we can equate the two equations:
8 = x^3
Solve for x:
x = 2
Step 2: Determine the radius of the shells.
- To find the radius of the shell, we need to consider a small vertical strip of thickness 'dx' at a given x-value. The distance from the x-axis to the curve y = x^3 is the radius of the shell at that x-value.
- In this case, the distance from the x-axis to the curve y = x^3 is given by y = x^3. So, the radius for each shell will be x^3.
Step 3: Determine the height of the shells.
- The height of each shell represents the length of the strip that is being revolved around the x-axis. In this case, the height will be the difference between the y-values of the upper and lower curves.
- The upper curve is y = 8, and the lower curve is y = x^3. So, the height of each shell will be given by 8 - x^3.
Step 4: Set up the integral.
- Now that we have the radius and height of each shell, we can set up the integral to calculate the volume.
- The formula for using the shell method is: V = 2π∫[a,b](radius * height) dx, where [a,b] represents the limits of integration.
- In this case, the limits of integration are from x = 0 to x = 2, as determined earlier.
- Therefore, the integral to calculate the volume will be: V = 2π∫[0,2](x^3 * (8 - x^3)) dx.
Step 5: Evaluate the integral.
- Now that we have set up the integral, we can proceed with evaluating it to find the volume.
- Using the power rule of integration and simplifying the equation, we get: V = 2π∫[0,2](8x^3 - x^6) dx.
- Evaluating this integral gives us the final volume of the solid generated.
I hope this step-by-step explanation helps you understand how to use the shell method to find the volume of the solid. If you have any further questions, feel free to ask!