DO NOW: Find the volume of the solid generated when the

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Section Volumes by Slicing

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Presentation transcript:

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. (2,25) Cross-section area: (0,5) Volume: f(x) x

The Washer Method Section 7.3c

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:

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

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:

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:

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

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: Cross section area: r

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 . Volume: r

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. Cross section radius: Cross section area: r Volume:

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!!! Cross section area: r R Volume:

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:

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:

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: