Find the area of the region bounded by the graphs of the equations. (Round your answer to three decimal places.) y = 3 sec ( 𝜋x/ 6) , x = 0, x = 2, y = 0

Question

Find the area of the region bounded by the graphs of the equations. Use a graphing utility to verify your result. (Round your answer to three decimal places.)

y = 3 sec(pi x /6), x = 0, x = 2, y = 0

Answer 1

This is simply just ∫₀² 3sec(πx/6)dx.

It is known that ∫sec(x)dx = ln(|sec(x)+tan(x)|) + C, so:

∫₀² 3sec(πx/6)dx

u = πx/6, du = (π/6)dx --> (18/π)du = 3dx

Converting our x-bounds to u-bounds, these are u= π(0)/6=0 to u=2π/6=π/3

(18/π)∫₀^π/3 sec(u)du

= (18/π)ln(|sec(u)+tan(u)|) [0,π/3]

= (18/π)ln(|sec(π/3)+tan(π/3)|) - (18/π)ln(|sec(0)+tan(0)|)

= (18/π)ln(2+√3) - (18/π)ln(1)

= (18/π)ln(2+√3) - (18/π)(0)

= (18/π)ln(2+√3)

≈ 7.546 (cubic units)

Hope this helped!