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Section 7.3 – Volume: Shell Method

Section 7.3 – Volume: Shell Method. White Board Challenge. Calculator. Calculate the volume of the solid obtained by rotating the region bounded by y = x 2 , x= 0, and y= 4 about x = -1 . We will now investigate another method to calculate this volume. Volume of a Shell.

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Section 7.3 – Volume: Shell Method

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  1. Section 7.3 – Volume: Shell Method

  2. White Board Challenge Calculator Calculate the volume of the solid obtained by rotating the region bounded by y = x2,x=0, and y=4 about x = -1. We will now investigate another method to calculate this volume.

  3. Volume of a Shell Consider the following cylindrical shell (formerly a washer): Imagine the circle in in the middle of the base area. Label the new radius. R The average of the radii is a new radius from the center of the base to the middle of the enclosed area. router rinner Thus, the circumference of the middle circle is… h Δr Also, the thickness of the shell is…

  4. Volume of a Shell The volume of the cylindrical shell is easier to see when it is flattened out: The cylindrical shell flattened out is a rectangular prism. The length of the base is… C = 2πR The height of the base is… h The height of the prism is… Δr Thus the volume of the prism is…

  5. Volumes of Solids of Revolution with Riemann Sums Let us rotate the region under y=f(x) from x=a to x=b about the y-axis. The resulting solid can be divided into thin concentric shells.

  6. Volumes of Solids of Revolution: Shell Method Opposite of Washer Method • Sketch the bounded region and the line of revolution. • If the line of revolution is horizontal, make sure the equations can easily be written in the x= form. If vertical, the equations must be in y=form. • Sketch a generic shell (a typical cross section). • Find the radius of the generic shell (perpendicular distance from the line revolution to the outer edge of the shell), the height of the shell (this length is perpendicular to the radius), and the thickness of the shell (this length is perpendicular to the height). • Integrate with the following formula: MAKE A HOOK:

  7. Example 1 Calculate the volume of the solid obtained by rotating the region bounded by y = x2,x=0, and y=4 about x= -1. Sketch a Graph Find the Boundaries/Intersections Radius = x – -1 = x+ 1 Thickness = dx We only need x>2 x Integrate the Volume of the Shell Height = 4 – x2 Make Generic Shell(s) Line of Rotation

  8. Example 2 Calculate the volume V of the solid obtained by rotating the region bounded by y = 5x – x2and y= 8 – 5x+ x2about the line y-axis. Sketch a Graph Find the Boundaries/Intersections Thickness = dx Integrate the Volume of Each Generic Washer Radius = x Line of Rotation Height = ( 5x– x2 ) – (8 – 5x + x2) Make Generic Shell(s)

  9. White Board Challenge Calculator Use the shell method to calculate the volume of the solid obtained by rotating the region bounded by y = x1/2 andy=0 over [0,4] about the x-axis. Height = 4 – y2 Thickness = dy Line of Rotation Radius = y

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