Find the volume generated by rotating the area between
y = cos( 3 x )
and the x axis from x = 0 to x = π/ 12 around the x axis
V = π∫cos(3x) dx from 0 to π/12
= π [ (1/3)sin(3x) ] from 0 to π/12
= π ( (1/3)sin(π/4) - (1/3)sin0 )
=(1/3) π (√2/2 - 0)
= √2 π/6
scrap that previous answer, I forgot to square it
V = π∫(cos(3x))^2 dx
We know
2cos^2 (3x) - 1= cos(6x)
cos^2 (3x) = 1/2 + (1/2)cos(6x)
then
V = π∫(1/2 + (1/2)cos(6x) ) form 0 to π/12
= π { x/2 + (1/12)sin(6x) ] from 0 to π/12
= π( π/24 + (1/12)sin(π/2) - 0 - 0 )
= π( π/24 + 0 )
= π^2 / 24
check my arithmetic
To find the volume generated by rotating the area between the curve y = cos(3x) and the x-axis from x = 0 to x = π/12 around the x-axis, we can use the method of cylindrical shells.
The volume generated by a cylindrical shell with height Δx at position x is given by the formula dV = 2πx * f(x) * Δx, where f(x) is the function that represents the curve.
To calculate the total volume, we need to integrate this expression over the given range. Let's go step by step:
Step 1: Determine the limits of integration.
The given range is from x = 0 to x = π/12.
Step 2: Set up the integral expression.
The integral expression to calculate the volume is:
V = ∫[0 to π/12] 2πx * f(x) dx
Step 3: Substitute the function f(x).
In this case, f(x) = cos(3x). Substituting this into the integral expression, we have:
V = ∫[0 to π/12] 2πx * cos(3x) dx
Step 4: Evaluate the integral.
To evaluate the integral, we need to use integration techniques or a symbolic calculator. Applying integration techniques, the solution will be:
V = [π * sin(3x) + 3/2 * x * cos(3x)] evaluated from 0 to π/12
Plugging in the limits of integration:
V = [π * sin(π/4) + 3/2 * π/12 * cos(π/4)] - [π * sin(0) + 3/2 * 0 * cos(0)]
Simplifying:
V = [π * (1/√2) + 3/2 * π/12 * (1/√2)] - 0
V = [π/√2 + π/√2] - 0
V = 2π/√2
So, the volume generated by rotating the area between the curve y = cos(3x) and the x-axis from x = 0 to x = π/12 around the x-axis is 2π/√2 cubic units.
To find the volume generated by rotating the area between the curve y = cos(3x) and the x-axis from x = 0 to x = π/12 around the x-axis, we can use the method of cylindrical shells.
The formula for calculating the volume using cylindrical shells is:
V = ∫[a, b] 2πx * f(x) * dx
Where:
- V is the volume
- a and b are the limits of integration
- x represents the axis of rotation
- f(x) is the function that represents the curve
In this case, our limits of integration are from x = 0 to x = π/12, and the function is f(x) = cos(3x). Plugging these values into the formula, we get:
V = ∫[0, π/12] 2πx * cos(3x) * dx
To solve this integral, we can use integration techniques. Let's go through the steps:
1. Integrate by parts:
Using the product rule, let u = x and dv = 2πcos(3x)dx.
Then, du = dx and v = (2π/3)sin(3x).
2. Apply the integration by parts formula:
∫ u * dv = uv - ∫ v * du
Using the values we obtained:
∫ 2πx * cos(3x) * dx = (2π/3)x * sin(3x) - ∫ (2π/3)sin(3x) * dx
3. Integrate the remaining integral:
To integrate ∫ (2π/3)sin(3x) * dx, we can use the substitution method. Let u = 3x, then du = 3dx, which gives us:
∫ (2π/3)sin(u) * (du/3)
= (2π/9)∫ sin(u) * du
= -(2π/9)cos(u) + C
4. Substitute back u = 3x:
= -(2π/9)cos(3x) + C
5. Evaluate the integral between the limits of integration:
Plugging the limits of integration (0 and π/12) into the formula, we get:
V = [-(2π/9)cos(3x)]ᴵ₀ˡⁿᵐⱼₜ(0,π/12)
= [-(2π/9)cos(3(π/12))] - [-(2π/9)cos(3(0))]
= -(2π/9)cos(π/4) + (2π/9)cos(0)
= -(2π/9)*(√2/2) + (2π/9)*1
= (π/9)(1 - √2)
Therefore, the volume generated by rotating the given area around the x-axis is (π/9)(1 - √2) cubic units.