Final Answer:
The volume of rotation for the region underneath the graph of y = x - x³ rotated about the x-axis is π/4.
Step-by-step explanation:
To find the volume of rotation, we need to integrate the area of the region over the range of the curve. The area of the region can be found by integrating the area of the rectangle whose height is the height of the curve at a given point and whose base is the x-axis.
The height of the curve at a point (x, y) is given by y = x - x³. Since we are rotating the curve about the x-axis, the height of the curve at a point (x, y) is the same as the height of the rectangle whose base is the x-axis and whose height is y. Therefore, the area of the region can be found by integrating the area of the rectangle over the range of the curve.
We can express the area of the rectangle as A = x * y, where x is the base and y is the height. To find the integral, we can substitute y = x - x³ into A = x * y and get:
A = ∫[x - x³] dx
Since we are integrating over the range of the curve, we can evaluate this integral from 0 to infinity.
Using the fundamental theorem of calculus, we have:
A = ∫[x - x³] dx = ∫[0 ∞] (x - x³) dx
We can evaluate this integral by substituting u = x - x³ and du = dx:
A = ∫[0 ∞] u du = ∫[0 ∞] (u² + 1/3) du
We can simplify this expression by factoring out the common factor of u²+ 1/3:
A = ∫[0 ∞] u² + 1/3 du = ∫[0 ∞] u^3 + 1/3 du
We can now evaluate this integral by substituting u = x³/3 and du = dx/3:
A = ∫[0 ∞] u³ + 1/3 du = ∫[0 ∞] (x^3/3)³+ 1/3 dx/3
We can simplify this expression by canceling out the 1/3 factor:
A = ∫[0 ∞] x³ dx/3 = π/4
Therefore, the volume of rotation for the region underneath the graph of y = x - x³ rotated about the x-axis is π/4.