Volumes of Solids of Rotation: The Disc Method

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Presentation transcript:

Volumes of Solids of Rotation: The Disc Method Animation of Rotation about the x-axis using the disc method

Rotations about the x-axis. You begin with a certain area under a curve on an interval from x = a to x = b Rotating that region about the x-axis generates a solid. The region could be “sliced” into cylindrical discs as shown in the diagram.

Total volume from x = a to x = b Radius of each cylindrical disc which is also the y-value or f(x). Formula for the Volume of a cylinder: Replace “r” with f(x) Replace “h” with x the width of each cylinder Total volume from x = a to x = b

AP only: General Formula for Rotations about any horizontal line using the disc method This formula should be used when there is no space between the line of rotation and the desired region. The radius, r, will be defined FROM your line of rotation. (f(x) is always defined from the x-axis)

The cross-section will be a triangular region in the first quadrant Let’s do an example: Find the volume of the solid formed when the region bounded by the line y = -x + 1 and the x and y axes is rotated about the x-axis. The cross-section will be a triangular region in the first quadrant

Setting up the integral: We know that: Substitute f(x) and limits of integration into equation Solve the integral!

Ex 2: Which of the following represents the correct integral for finding the volume of the solid formed by rotating the region bounded by y = x2 and the x-axis about the x-axis on the interval [0, 1]? No, this one gives you area! Close, but something is missing! Yes, this one is correct!

Volumes of Solids of Rotation: The Washer Method To be used when there is a space between the line of rotation and the shaded area.

General Formula for Rotations about the horizontal lines using the washer method

Rotate the region bounded by y = 8-x2 and y =x2 from [-2,2] about the x-axis

Example 1: Washer Method Find the volume of the solid obtained by rotating about the x- axis the region bounded by the graphs of f(x) = x1/2 and g(x) = x2

Example 2: Washer Method Find the volume of the solid obtained by revolving about the x- axis the region bounded by the curves f(x) = sin x and g(x) = cos x on [0, /2]

VOLUMES: DISC AND WASHER METHODS REVOLVING ABOUT THE Y-AXIS

Formula limits on y-axis solve equation for x to get f(y) Remember: boundary is now between the curve and the y-axis Note: these will not always be functions.

Example 1: Find the volume of the region rotated about the y-axis bounded by the curve and the y-axis

Example 2: Find the volume of the region in the first quadrant bounded by the graph of and the coordinate axes by rotating about the y-axis.

Between two curves – Example 1 Find the volume between the graphs of x = 3, y = 0, and y = 2x rotated about the y-axis. farthest from y-axis – closest to y-axis

Example 2: Find the volume found by rotating the region bounded by and about the y-axis.