Final answer:
To find the volume of the solid obtained by rotating the region bounded by the curves 2x = y², x = 0, y = 5 about the y-axis, we can use the method of cylindrical shells. The volume can be calculated by integrating the expression πy³dy over the range of y values that define the region. The resulting volume is (625/4)π.
Step-by-step explanation:
To find the volume V of the solid obtained by rotating the region bounded by the curves 2x = y², x = 0, y = 5 about the y-axis, we can use the method of cylindrical shells. The volume of each shell can be calculated as 2πrhΔx, where r is the distance from the y-axis to the curve, h is the height of the shell, and Δx is the width of the shell. In this case, the distance from the y-axis to the curve is x = y²/2, the height of the shell is y, and the width of the shell can be represented as dy (since we are rotating about the y-axis).
Therefore, the volume of each shell is 2π(y²/2)(y)dy = πy³dy. To find the total volume, we integrate this expression over the range of y values that define the region bounded by the curves. Since the region is bounded by y = 0 and y = 5, the integral becomes ∫(0 to 5)πy³dy. Evaluating this integral gives the volume V = π[(5^4)/4] - π[(0^4)/4] = (625/4 - 0/4)π = (625/4)π.