The region in the first quadrant bounded by the curve 𝑦 = cos 𝑥 𝑎𝑛𝑑 𝑡ℎ𝑒 𝑙𝑖𝑛𝑒𝑠 𝑦 = 1 𝑎𝑛𝑑 𝑥 =1/2𝜋 𝑖𝑠 𝑟𝑒𝑣𝑜𝑙𝑣𝑒𝑑 𝑎𝑏𝑜𝑢𝑡 𝑡ℎ𝑒 𝑡ℎ𝑒 𝑙𝑖𝑛𝑒 𝑥 =1/2𝜋. Find the volume of the solid generated.

Question

The region in the first quadrant bounded by the curve 𝑦 = cos 𝑥 𝑎𝑛𝑑 𝑡ℎ𝑒 𝑙𝑖𝑛𝑒𝑠 𝑦 = 1 𝑎𝑛𝑑 𝑥 =1/2𝜋 𝑖𝑠 𝑟𝑒𝑣𝑜𝑙𝑣𝑒𝑑 𝑎𝑏𝑜𝑢𝑡 𝑡ℎ𝑒 𝑡ℎ𝑒 𝑙𝑖𝑛𝑒 𝑥 =1/2𝜋. Find the volume of the solid generated.

Answer 1

using cylinders of thickness dx, we have

v = ∫[0,π/2] 2πrh dx
where r = π/2 - x and h = 1-cosx
v = ∫[0,π/2] 2π(π/2 - x)(1-cosx) dx = π/4 (π^2 - 8) ≈ 1.4684

Answer 2

To find the volume of the solid generated by rotating the region bounded by the curves 𝑦 = cos(𝑥), 𝑦 = 1, and 𝑥 = 1/2𝜋 in the first quadrant around the 𝑥-axis, we can use the method of cylindrical shells.

1. Draw the graph of the given curves in the first quadrant and identify the region that needs to be rotated.

2. Determine the limits of integration. Since we are rotating around the 𝑥-axis, the limits of integration will be from 𝑥 = 0 to 𝑥 = 1/2𝜋.

3. Set up the integral. The volume of a cylindrical shell is given by the formula:

𝑉 = ∫[𝑎,𝑏] 2𝜋𝑥(𝑓(𝑥) - 𝑔(𝑥)) 𝑑𝑥

where 𝑦 = 𝑓(𝑥) is the upper function and 𝑦 = 𝑔(𝑥) is the lower function.

In this case, the upper function 𝑦 = 𝑓(𝑥) is 𝑦 = 1 and the lower function 𝑦 = 𝑔(𝑥) is 𝑦 = cos(𝑥). Therefore, the integral becomes:

𝑉 = ∫[0,1/2𝜋] 2𝜋𝑥(1 - cos(𝑥)) 𝑑𝑥

4. Evaluate the integral. This can be done by either using antiderivatives or by using numerical integration methods.

Once the integral is evaluated, you will obtain the volume of the solid generated by rotating the region in the first quadrant.

Note: If you are using a software or calculator to evaluate the integral, make sure to use the appropriate units for the volume.