Final answer:
To find the volume of the solid obtained when the region under the curve y = 15 arcsin(x), x ≥ 0, is rotated about the y-axis, we can use the method of cylindrical shells. The formula for the volume is V = 2π ∫(x∙f(x)dx), where x is the variable of integration and f(x) is the equation of the curve.
Step-by-step explanation:
To find the volume of the solid obtained when the region under the curve y = 15 arcsin(x), x ≥ 0, is rotated about the y-axis, we can use the method of cylindrical shells. The formula for the volume using cylindrical shells is V = 2π ∫(x∙f(x)dx), where x is the variable of integration and f(x) is the equation of the curve. In this case, f(x) = 15 arcsin(x). So the volume is given by V = 2π ∫(x∙15 arcsin(x)dx), where the limits of integration are from 0 to 1.
Using the table of integrals, we can find that the integral of x∙sin^(-1)(x)dx is x^2∙sin^(-1)(x) + sqrt(1-x^2)/2 + C.
Therefore, the volume of the solid is V = 2π [(1^2∙sin^(-1)(1) + sqrt(1-1^2)/2) - (0^2∙sin^(-1)(0) + sqrt(1-0^2)/2)].