Volume of a Function

The volume of the solid revolution generated by rotating the curve f(x) around the x-axis and bounded by x = a and x = b, is given by:

Volume of a Function

Examples

1. Find the volume of a truncated cone that is generated by the rotation around the line y = 6 − x and bounded by the lines y = 0, x = 0, x = 4.

Volume of a Function

Volume of a Function Solution

2. Calculate the volume generated by y = sin x when rotated about the x-axis.

Volume of a Function

Volume Integral

Integral Solución

3. Find the volume of the solid revolution generated by rotating the function f(x) = 1/2 + cos x (bounded by the x-axis and the lines x = 0 and x = π) around the x-axis.

Volume of a Function Integral

Volume of a Function Solution

4. Find the volume generated by the circle x2 + y2 − 4x = −3 rotating about the x-axis.

Equation

Equation

The center of the circumference is C(0, 1) and radius r = 1.

x-intercepts:

x-intercepts

Semi-Circle

Volume of a Function

Volume of a Function Solution

5. Calculate the volume generated by rotating around the x-axis, the site bounded by the graphs of y = 2x − x2 and y = −x + 2.

The points of intersection between the parabola and the straight line:

Points on the Curve

Volume of a Function

The parabola is above the straight line in the interval of integration.

Volume of a Function

Volume of a Function Solution

6. Calculate the volume generated by rotating the site bounded by the graphs of y = 6x − x2 and y = x around the x-axis .

The points of intersection:

Points of Intersection

Volume of a Function

The parabola is above the straight line in the interval of integration.

Volume of a Function

7. Calculate the volume generated by a triangle of vertices A(3, 0), B(6, 3), C(8, 0) that rotates 360º around the x-axis.

The equation of the straight line that passes through AB:

Equation

The equation of the straight line that passes through BC:

Equation

Triangle

Volume of a Function

Volume of a Function Operations

Volume of a Function Operations

Volume of a Function Solution

8.Find the volume of the figure generated by rotating the ellipse Equation of an Ellipse around the x-axis.

Ellipse

Equation

As the ellipse is a symmetrical curve, the volume is 2 times the volume generated by the arc Arc of an Ellipse between x = 0 and x = a.

Volume of a Function

Volume of a Function Solution