Final Answer:
The volume of the solid obtained by rotating the region underneath the graph of y = 2x³ about the y-axis over the interval [0, 2] is 32/5π cubic units.
Step-by-step explanation:
To find the volume of the solid of revolution, integrate the expression πf(x)² over the given interval [0, 2], where f(x) = 2x³ represents the function defining the region. The integral is calculated as follows:
V = ∫[0 to 2] π(2x³)² dx
Simplify the expression by squaring and multiplying:
V = ∫[0 to 2] π * 4x⁶ dx
Integrate with respect to x:
V = π [4/7 * x⁷] from 0 to 2
Evaluate the expression at the upper and lower limits:
V = π [4/7 * (2)⁷ - 4/7 * (0)⁷]
Simplify further:
V = π * [4/7 * 128]
V = 32/5π
Therefore, the volume of the solid obtained by rotating the region underneath the graph of y = 2x³ about the y-axis over the interval [0, 2] is 32/5π cubic units.