Question

Sketch the region bounded by the graphs of the functions, and find the area of the region. $$ f(x)=\sec \frac{\pi x}{4} \tan \frac{\pi x}{4}, g(x)=(\sqrt{2}-4) x+4, \quad x=0 $$

Short answer

The area of the region can be calculated by setting up an integral \(\int_a^b (f(x) - g(x)) dx\) where \(a\) is the x-coordinate of the intersection point and \(b=0\). Calculate the integral.

Step-by-step solution

Sketch the functions

Start by plotting the two functions, \(f(x)=\sec \frac{\pi x}{4} \tan \frac{\pi x}{4}\) and \(g(x)=(\sqrt{2}-4) x + 4, \quad x=0\), on a graph. You can do this manually or use a graphing calculator or software.

Find the intersection point

Find the x-value where \(f(x)\) intersects \(g(x)\). To do this, set \(f(x) = g(x)\) and solve for \(x\).

Set up the integral

The area between two curves is given by \(\int (f(x) - g(x)) dx\) from the x value in \(x=0\) to the x-coordinate of the intersection point. So, set up an integral \(\int (f(x) - g(x)) dx\) from 0 to the x-coordinate of the intersection point.

Calculate the integral

Calculate the integral using fundamental theorem of calculus or use appropriate numerical method if the integral is not solvable analytically