Chapters
Exercise 1
Find the volume of the solid obtained by rotating the region bounded by y = sen x, x = 0 and x = π and the x-axis about the x-axis.
Exercise 2
Calculate the volume of the cylinder generated by the rectangle bounded by straight lines y = 2, x = 1, x = 4, and the x-axis to rotate around the x-axis.
Exercise 3
Find the volume of the truncated cone generated by the trapezoid contained by the x-axis, the lines y = x + 2, x = 4 and x = 10, which turns around the x-axis.
Exercise 4
Find the volume generated by rotating the region bounded by y = 2x − x² and y = −x + 2 around the x-axis.
Exercise 5
Find the volume generated by rotating the region bounded by y²/8 = x and x = 2, around the y-axis.
Exercise 6
Calculate the volume of a sphere of radius r.
Exercise 7
Find the volume of the ellipsoid generated by the ellipse 16x² + 25y² = 400 and turning:
1 Around its major axis.
2 Around its minor axis.
Solution of exercise 1
Find the volume of the solid obtained by rotating the region bounded by y = sen x, x = 0 and x = π and the x-axis about the x-axis.
y = sen xx = 0x = π
Solution of exercise 2
Calculate the volume of the cylinder generated by the rectangle bounded by straight lines y = 2, x = 1, x = 4, and the x-axis to rotate around the x-axis.
Solution of exercise 3
Find the volume of the truncated cone generated by the trapezoid contained by the x-axis, the lines y = x + 2, x = 4 and x = 10, which turns around the x-axis.
Solution of exercise 4
Find the volume generated by rotating the region bounded by y = 2x − x² and y = −x + 2 around the x-axis.
The points of intersection between the parabola and the line:
The parable is above the line in the interval of integration.
Solution of exercise 5
Find the volume generated by rotating the region bounded by 
and x = 2, around the y-axis.
As it turns about the y-axis, apply:
Since the parabola is symmetrical about the x-axis, the volume is equal to two times the volume generated between y = 0 and y = 4.
Solution of exercise 6
Calculate the volume of a sphere of radius r.
Start from the equation of the circumference x² + y² = r².
Turning a semicircle around the x-axis gives a sphere.
Solution of exercise 7
Find the volume of the ellipsoid generated by the ellipse 16x² + 25y² = 400 and turning:
1 Around its major axis.
2 Around its minor axis.
As the ellipse is symmetric about two axes, the volume is double the portion generated by the ellipse in the first quadrant in both cases.
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