Find the volume generated by rotating about the x axis the area bounded by x=0, y=2sinxcosx,, y=2cosx, and x=4, if 0≤x≤pi/2.
I don't get the graph.
You're right. The boundaries are not clear to me either.
http://www.wolframalpha.com/input/?i=plot+y%3D2sinxcosx%2C+y%3D2cosx
y=2sinxcosx=sin2x
To find the volume generated by rotating the given area about the x-axis, we first need to visualize the graph of the curves involved.
To better understand the graph, let's examine each curve separately:
1. x = 0: This is simply a vertical line at x = 0.
2. y = 2sin(x)cos(x): This curve is a sinusoidal function with amplitude 2 and frequency dependent on x.
3. y = 2cos(x): This curve is a cosine function with amplitude 2 and a period of 2π.
4. x = π/2: This is a vertical line at x = π/2.
5. x = 4: This is another vertical line at x = 4, which is outside the given range.
Now let's focus on the interval of interest, 0 ≤ x ≤ π/2.
Start by plotting the curves individually on a graphing calculator or software:
1. Plot y = 2sin(x)cos(x) and y = 2cos(x) together. This will give you an idea of how the sinusoidal curve and the cosine curve interact with each other within the given interval.
2. Add the vertical lines at x = 0 and x = π/2 to complete the graph.
As a result, you should see the area bounded by the curves within the given interval. This is the region that will be rotated about the x-axis to find the volume.
Please note that the graph may look complex due to the trigonometric functions involved, but visualizing it should help you better understand the problem.