Find the volume generated if the region bounded by y = sin 2x and y = cos x, π/6 ≤� x ≤ �π/2
is revolved about the line y = 2. Use the method of washers.
To find the volume generated by revolving the region about the line y = 2 using the method of washers, we need to integrate the cross-sectional areas of the washers.
Step 1: Determine the bounds of integration
Given that the region is bounded by y = sin 2x and y = cos x, and π/6 ≤ x ≤ π/2, we need to find the values of y that correspond to the upper and lower boundaries of the region. To do this, we'll equate the two functions:
sin 2x = cos x
Rearranging the equation:
sin 2x - cos x = 0
Now, solve for x by using trigonometric identities or solving graphically. We find two solutions within the given range: x1 ≈ 0.927 and x2 ≈ 1.309.
Next, plug these values into the equations to find the corresponding y-values:
For x1 ≈ 0.927:
y1 = sin(2 * 0.927) ≈ 0.80 (upper bound)
y2 = cos(0.927) ≈ 0.60 (lower bound)
For x2 ≈ 1.309:
y3 = sin(2 * 1.309) ≈ 0.99 (upper bound)
y4 = cos(1.309) ≈ 0.24 (lower bound)
Step 2: Set up the integral
To find the volume using the method of washers, we integrate the cross-sectional area formula π * [(outer radius)² - (inner radius)²] with respect to x.
The outer radius (R) is the distance from the axis of revolution (y = 2) to the upper curve (y = sin 2x). Therefore, R = 2 - y.
The inner radius (r) is the distance from the axis of revolution to the lower curve (y = cos x). Hence, r = 2 - y.
The cross-sectional area of each washer is given by:
A(x) = π * [R(x)² - r(x)²]
Step 3: Evaluate the integral
The volume can be computed as:
V = ∫[π * (2 - y)² - (2 - y)²] dx, within the interval π/6 ≤ x ≤ π/2
We express y in terms of x using the upper and lower bounds determined earlier:
V = ∫[π * (2 - sin 2x)² - (2 - cos x)²] dx, π/6 ≤ x ≤ π/2
Calculating this integral will yield the volume of the solid generated by revolving the region about the line y = 2.