Finding the Volume of Revolution
To find the volume of the solid formed by revolving the region bounded by the curves y = x³ + x² and y = 1 around the y-axis, we can use the method of cylindrical shells.
1. Identify the Region
First, let’s determine the intersection points of the two curves.
Set:
x³ + x² = 1
This can be rearranged to:
x³ + x² – 1 = 0
To find the roots of this polynomial, we can use numerical methods or graphing:
- By plotting the graph, we find that it intersects at approximately x ≈ 0.68 and x = -1.
2. Setting Up the Integral
Using the shell method, the volume V can be calculated using the formula:
V = 2π ∫[a to b] (radius)(height) dy
In this case, the radius is x and the height is given by 1 – (x³ + x²).
To convert the bounds in terms of y, we solve for x in terms of y from the equation y = x³ + x². This is non-trivial algebraically, so we will keep the variables in terms of x.
3. The Integral
The volume is then expressed as:
V = 2π ∫[0 to 0.68] x(1 – (x³ + x²)) dx
This integral can be split into two parts:
V = 2π ∫[0 to 0.68] (x – x⁴ – x³) dx
4. Calculating the Integral
We can now compute the integral:
V = 2π [0.5x² – (1/5)x⁵ – (1/4)x⁴] from 0 to 0.68
Evaluating the bounds:
At x = 0.68:
V = 2π [0.5(0.68)² – (1/5)(0.68)⁵ – (1/4)(0.68)⁴]
At x = 0:
V = 0
5. Final Volume Calculation
Plugging in the values will provide the final volume:
Calculate the numeric value of the integral to find:
V ≈ 0.4724π
Conclusion
The volume of the solid formed by revolving the specified region around the y-axis is approximately:
V ≈ 1.484 cubic units