VOLUMES.

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Presentation transcript:

VOLUMES

1 Disk cross-section x VOLUMES If the cross-section is a disk, we find the radius of the disk (in terms of x ) and use Rotating axis animation java

1 Disk cross-section x step1 step2 step3 step6 step4 step5 VOLUMES Intersection point between L, curve step1 Graph and Identify the region step2 Draw a line perpendicular (L) to the rotating line at the point x Intersection point between L, rotating axis step3 Find the radius r of the disk in terms of x step6 Now the cross section Area is step4 The volume is given by Specify the values of x step5

VOLUMES

VOLUMES

Volume = Area of the base X height VOLUMES Volume = Area of the base X height

VOLUMES

2 washer cross-section x VOLUMES If the cross-section is a washer ,we find the inner radius and outer radius

2 washer cross-section x step1 step2 step3 step6 step4 step5 VOLUMES Intersection point between L, boundry Graph and Identify the region step1 Draw a line perpendicular to the rotating line at the point x Intersection point between L, rotation axis step2 Find the radius r(out) r(in) of the washer in terms of x Intersection point between L, boundary step3 step6 Now the cross section Area is step4 The volume is given by Specify the values of x step5

VOLUMES T-102

VOLUMES Example: Find the volume of the solid obtained by rotating the region enclosed by the curves y=x and y=x^2 about the line y=2 . Find the volume of the resulting solid.

3 Disk cross-section y VOLUMES If the cross-section is a disk, we find the radius of the disk (in terms of y ) and use

3 Disk cross-section y step1 step2 step2 step3 step6 step4 step5 VOLUMES 3 Disk cross-section y Graph and Identify the region step1 Rewrite all curves as x = in terms of y step2 Draw a line perpendicular to the rotating line at the point y step2 Find the radius r of the disk in terms of y step3 step6 Now the cross section Area is step4 The volume is given by Specify the values of y step5

4 washer cross-section y VOLUMES Example: The region enclosed by the curves y=x and y=x^2 is rotated about the line x= -1 . Find the volume of the resulting solid.

4 washer cross-section y step1 step2 step2 step3 step6 step4 step5 VOLUMES 4 washer cross-section y Graph and Identify the region step1 Rewrite all curves as x = in terms of y step2 Draw a line perpendicular to the rotating line at the point y step2 Find the radius r(out) and r(in) of the washer in terms of y step3 step6 Now the cross section Area is step4 The volume is given by Specify the values of y step5

Cross—section is WASHER VOLUMES SUMMARY: The solids in all previous examples are all called solids of revolution because they are obtained by revolving a region about a line. Rotated by a line parallel to x-axis ( y=c) solids of revolution Rotated by a line parallel to y-axis ( x=c) NOTE: The cross section is perpendicular to the rotating line Cross-section is DISK solids of revolution Cross—section is WASHER

Cross—section is WASHER VOLUMES BY CYLINDRICAL SHELLS Remarks rotating line Parallel to x-axis CYLINDRICAL SHELLS (6.2) rotating line Parallel to y-axis Remarks rotating line Parallel to x-axis DESK(6.1) rotating line Parallel to y-axis Cross-section is DISK Cross—section is WASHER SHELL Method

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VOLUMES T-102

VOLUMES BY CYLINDRICAL SHELLS T-131

not solids of revolution VOLUMES solids of revolution not solids of revolution

VOLUMES 3 not solids of revolution The solids in all previous examples are all called solids of revolution because they are obtained by revolving a region about a line. We now consider the volumes of solids that are not solids of revolution. 3 side

VOLUMES T-102

VOLUMES T-122

VOLUMES T-092

VOLUMES