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DO NOW: Find the volume of the solid generated when the region in the first quadrant bounded by the given curve and line is revolved about the x-axis. (0,5) (2,25) f(x) x Cross-section area: Volume:

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SECTION 7.3C The Washer Method

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The region in the first quadrant enclosed by the y-axis and the graphs of y = cos(x) and y = sin(x) is revolved about the x-axis to form a solid. Find its volume. Graph the region…and visualize the solid… Each cross section perpendicular to the axis of revolution is a washer, a circular region with a circular region cut from its center: R r Area of a washer:

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The region in the first quadrant enclosed by the y-axis and the graphs of y = cos(x) and y = sin(x) is revolved about the x-axis to form a solid. Find its volume. The outer and inner radii are the y values of our two functions!!! Cross section area: Volume: units cubed

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Guided Practice Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis. Cross section area: Volume:

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Guided Practice Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis. Cross section area: Volume:

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Guided Practice Find the volume of the solid generated by revolving the given region about the y-axis. Cross section area: Volume: The region bounded above by the curve and below by the line.

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Guided Practice – Other Lines of Revolution!!! Find the volume of the solid generated by revolving the region in the first quadrant bounded above by the line, below by the curve,, and on the left by the y-axis, about the line. Cross section radius: r Cross section area:

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Guided Practice – Other Lines of Revolution!!! Find the volume of the solid generated by revolving the region in the first quadrant bounded above by the line, below by the curve,, and on the left by the y-axis, about the line. r Volume:

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Guided Practice – Other Lines of Revolution!!! Find the volume of the solid generated by revolving the triangular region bounded by the lines y = 2x, y = 0, and x = 1 about (a) the line x = 1. r Cross section radius: Cross section area: Volume:

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Guided Practice – Other Lines of Revolution!!! Find the volume of the solid generated by revolving the triangular region bounded by the lines y = 2x, y = 0, and x = 1 about (b) the line x = 2. Washers!!! R r Cross section area: Volume:

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Guided Practice – Other Lines of Revolution!!! Find the volume of the solid generated by revolving the region bounded by the parabola and the line about (a) the line y = 1. Cross section: Volume:

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Guided Practice – Other Lines of Revolution!!! Find the volume of the solid generated by revolving the region bounded by the parabola and the line about (b) the line y = 2. Washers: Volume:

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Guided Practice – Other Lines of Revolution!!! Find the volume of the solid generated by revolving the region bounded by the parabola and the line about (c) the line y = –1. Washers: Volume:

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