Final answer:
To find the area of the region bounded by y = 256√x and y = 32x², we need to find the points of intersection of the two curves and calculate the definite integral.
Step-by-step explanation:
To find the area of the region bounded by y = 256√x and y = 32x², we need to find the points of intersection of the two curves. Setting the two equations equal to each other, we have:
256√x = 32x²
Divide both sides by 32x² to get:
8√x = x²
Square both sides of the equation:
64x = x&sup4;
Rearrange the equation:
x&sup4; - 64x = 0
Factor out x:
x(x³ - 64) = 0
From this, we can see that x = 0 and x = 4 are the two points of intersection.
To find the area of the region between the two curves, we need to calculate the definite integral of the difference between the two equations from x = 0 to x = 4:
Area = ∫(256√x - 32x²) dx, evaluated from 0 to 4
After calculating the integral, we find that the area of the region is 2048 square units.