Volume of Function Word Problems

Solutions

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.

2Calculate 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.

3Find 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.

4 Find the volume generated by rotating the region bounded by y = 2x − x2 and y = −x + 2 around the x-axis.

5 Find the volume generated by rotating the region bounded by y2/8 = x and x = 2, around the y-axis.

6Calculate the volume of a sphere of radius r.

7Find the volume of the ellipsoid generated by the ellipse 16x2 + 25y2 = 400 and turning:

1 Around its major axis.

2 Around its minor axis.


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

Volume of Function Solution


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.

Volume of Function Solution


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.

Volume of Function Operations

Volume of Function Solution


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:

Volume of Function Operations

Line and Parabola

The parable is above the line in the interval of integration.

Volume of Function Operations

Volume of Function Solution


5

Find the volume generated by rotating the region bounded by y2/8 = x and x = 2, around the y-axis.

As it turns about the y-axis, apply:

Volume of Function Operations

Horizontal Parabola

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.

Volume of Function Solution


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.

Sphere

Volume of Function Operations

Volume of Function Solution


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.

Ellipse

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.

Volume of Function Operations

Volume of Function Operations

Volume of Function Operations

Volume of Function Solution