use the cylindrical shell method to find the volume of the solid obtained by rotating the region bounded by the curve y=sin(7x^2) and the x-axis, 0<x<sqrt(pi/7), about the y-axis
AAAaannndd the bot gets it wrong yet again!
This step is wrong: V = π/7 ∫(0 to pi) sin(u) du/7
and integration by parts is unnecessary; in fact, useless
∫[0,√(π/7)] 2πxsin(7x^2) dx = π/7
To use the cylindrical shell method, we need to integrate the circumference of cylindrical "shells" formed by rotating a vertical slice of the region about the y-axis.
The radius of each cylindrical shell is the distance from the y-axis to the curve, which is y = sin(7x^2).
The height of each cylindrical shell is the width of the slice, which is dx.
The limits of integration are 0 and sqrt(pi/7), the bounds of the region in the x-axis.
Therefore, the volume can be calculated as follows:
V = ∫(0 to sqrt(pi/7)) 2πx sin(7x^2) dx
Using u-substitution with u = 7x^2 and du/dx = 14x, we can simplify this to:
V = π/7 ∫(0 to pi) sin(u) du/7
Using integration by parts with u = sin(u) and dv = du/7 gives:
V = π/7 [-cos(u)/49] (0 to pi)
Plugging in the limits of integration gives:
V = π/7 [1/49 + 1/49] = π/245
Thus, the volume of the solid obtained by rotating the region about the y-axis is π/245 cubic units.