Let R be the region in the first quadrant that is enclosed by the graph of y = tanx, the x-axis, and the line x = π/3

Question

h. Find the area of R
i. Find the volume of the solid formed by revolving R about the x-axis

Answer 1

what's the problem?

A = ∫[0,pi/3] tanx dx

integrating tanx is easy, since
tanx = sinx/cosx = -d(cosx)/cosx
∫tanx dx = -ln(cosx)

A = -ln(cosx) [0,pi/3]
= -ln(cos pi/3) + ln(cos 0)
= -ln(1/2) + ln(1)
= ln2

v = ∫[0,pi/3] pi*r^2 dx
= pi*∫[0,pi/3] tan^2x dx
= pi*∫[0,pi/3](sec^2x - 1) dx
= pi*(tanx - x) [0,pi/3]
. . .