Area of Region, Volume of Solid, Volume of Solid Generated

Question

Let R be the region bound by the equations 𝑦 = 1 + cos(𝑥) and 𝑦 = csc(𝑥) in the first quadrant on the interval 0 ≤ x < π. a) Write, but do not solve, an equation involving integral expressions whose solution is the area of the region R. b) Write, but do not solve, an equation involving integral expressions whose solution is the volume of the solid generated when R is revolved around the x-axis. c) Write, but do not solve, an equation involving integral expressions whose solution is the volume of the solid generated when R is revolved around the line x = –1.

Answer 1

a) 1 + cosx = 1/sinx, sinx + sinxcosx = 1; x = 0.574 = a;or x = 1.5708 = b (we use TI-84)

Area A = ∫_a^b(1 + cosx - cscx)dx;

b) Volume V = π∫_a^b[(1 + cosx)² - csc²x]dx.

c) Volume V = 2π∫_a^b(x + 1)(1 + cosx - cscx)dx