Problems

In Problems 1-12 find the derivatives.

In Problems 16-25 evaluate the limit.

52            The region bounded by the curve y = 1/√x, 1 ≤ x ≤ 4, is rotated about the x-axis. Find the volume of the solid of revolution.

53            Find the volume generated by rotating the region under the curve y = ln x, 1 ≤ x ≤ e, about (a) the x-axis, (b) the y-axis.

54            Find the volume generated by rotating the region under the curve y = - ln x, 0 < x ≤ 1, about (a) the x-axis, (b) the y-axis.

55            Find the length of the curve y = ln x, 1 ≤ x ≤ e.

56            Find the surface area generated by rotating the curve y = ln x, 0 ≤ x ≤ 1, about the y-axis.

In Problems 29-51 evaluate the integral.

57            The inverse hyperbolic sine is defined by

Show that this is the inverse of the hyperbolic sine function by solving the equation below for y:

58            Show that

59            Show that

is the inverse function of tanh y, and that d(arctanh x) = 1/(1 - x²).