Find the volume of the solid obtained by rotating the region in the first quadrant enclosed by the curves y=tanx and y= 2​ cosx, for 0≤x≤ 2π​ , about the line y=−1

To find the volume of the solid obtained by rotating the region in the first quadrant enclosed by the curves y=tanx and y=2cosx, for 0≤x≤2π, about the line y=-1, we can use the method of cylindrical shells.

Step 1: Identify the region to be rotated The region is bounded by the curves y=tanx and y=2cosx, for 0≤x≤2π.

Step 2: Set up the integral for the volume of the solid The volume V of the solid obtained by rotating the region about the line y=-1 is given by the integral:

V = ∫_a^b 2πrh dx

where r is the distance from the line y=-1 to the point (x,y) on the curve, and h is the height of the cylindrical shell at (x,y).

In this case, r = y + 1 (since the line y=-1 is 1 unit below the x-axis), and h = 2cosx - tanx (since y=2cosx is above y=tanx for 0≤x≤2π).

So the integral becomes:

V = ∫_0^2π 2π(y + 1)(2cosx - tanx) dx

Step 3: Evaluate the integral This is a somewhat complicated integral to evaluate. You would likely need to use integration techniques such as integration by parts or trigonometric substitution.

Once you evaluate the integral, you will have the volume of the solid.