1. The curve has an equation y = ex. Compute the area bounded by the curve from x = 0 to x = 1.
2. The loop of the curve has an equation of y2 = x(1 – x)2. Find the area enclosed by the loop of the curve.
3. Given the area in the first quadrant bounded by y2 = x, the line x = 4 and the x-axis. What is the volume generated when this area is revolved about the y-axis?
4. The region in the first quadrant which is bounded by the curve y2 = 4x, and the lines x = 4 and y = 0, is revolved about the x-axis. Locate the centroid of the resulting solid of revolution.
5. The region in the first quadrant, which is bounded by the curve x2 = 4y, the line x = 4, is revolved about the line x = 4. Locate the centroid of the resulting solid of revolution.
Solution.
1.
2.
Find for this we do replacement
Then
So,
Therefore,
3.
4.
The volume of the body V formed by rotation around the axis Ox of the figure , where y1(x) and y2(x) are continuous non-negative functions, is equal to a certain interval from the difference of the square of the functions yi(x) by the variable x:
We will have
In this case, the centroid lies on the x-axis. The formula to get the centroid of the figure is:
From here, centroid
5.
The volume of the body V formed by rotation around the line x=m of the figure , where x(y) is a unique continuous function equal to the definite integral calculated by the formula
In this case, the centroid lies on the y-axis. The formula to get the centroid of the figure is
From here
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