(Arc lengths, Exercises 7.3) Find the lengths of the given curves.

(a) y =

12

+

–

from x = 1 to x = 4.

(b) y = x2

In x

8

from x = 1 to x = 2

(Surface area, Exercises 7.3)

(a) Find the area of the surface obtained by rotating y = sinx (0 lt; lt; lt; lt;)

about the r-axis.

(b) Find the surface area of the torus (doughnut) obtained by rotating the circle

(x – b)2 + y2 = a2 about the y-axis.Math