Find the volume of the solid whose base is the region bounded between the curve y=sec x and the x-axis from x=pi/4 to x=pi/3 and whose cross sections taken perpendicular to the x-axis are squares.
To find the volume of the solid with square cross sections, we need to integrate the areas of the squares over the given interval.
First, let's find the side length of each square.
The area of a square is given by the formula:
Area = side length * side length
Since the cross sections are squares, the side length will be equal to the difference between the curve y=sec(x) and the x-axis, which is sec(x). Therefore, the area of each square is (sec(x))².
Next, we need to integrate the area of each square over the given interval [π/4, π/3].
The volume of the solid can be calculated using the integral:
Volume = ∫[π/4, π/3] (sec(x))² dx
To evaluate the integral, we can use a symbolic math tool or a numerical integration method, such as numerical approximation or numerical integration software.
Using a symbolic math tool, the integral can be evaluated as follows:
Volume = ∫[π/4, π/3] (sec(x))² dx
= [tan(x)]|[π/4, π/3]
= tan(π/3) - tan(π/4)
The final step is to evaluate tan(π/3) and tan(π/4), and subtract the results.
tan(π/3) = √3
tan(π/4) = 1
Therefore, the volume of the solid is √3 - 1 cubic units.