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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