Calculus

Volume of Revolution

Volume of Revolution - Shell Method

         

The above diagram shows the points R=(7,17)R=(7,\frac{1}{7}) and P=(14,114),P=(14,\frac{1}{14}), which lie on the curve y=1x,y=\frac{1}{x}, point Q=(0,17)Q = (0, \frac{1}{7}) on the yy-axis, and points A=(7,0)A = (7,0) and B=(14,0)B = (14,0) on the xx-axis. If SS is the solid obtained by rotating the region RR bounded by OQ,QR,RP^,PB, and OB\overline{OQ}, \overline{QR}, \widehat{RP}, \overline{PB}, \text{ and } \overline{OB} about the axis of rotation x=7x=-7 and the volume of SS is απ,\alpha \pi, what is α?\alpha?

Note: The above diagram is not drawn to scale.

Let SS be the solid obtained by rotating the region RR bounded by y=x2 y = x^2 and y=x y = \sqrt{x} about the line x=10x=-10 . If the volume of SS is απ,\alpha \pi, what is α?\alpha?

Let SS be the solid obtained by rotating the region RR bounded by y=x9,x=15, y = x-9, x = 15, and the xx-axis about the axis of rotation x=19.x=19 . If the volume of SS is απ,\alpha \pi, what is α?\alpha?

Let SS be the solid obtained by rotating the region RR bounded by y=x3,x=1,x=5,y = x^3, x = 1, x = 5, and the xx-axis about the line x=8x=8 . If 1010 times the volume of SS is απ,\alpha \pi , what is α?\alpha?

Let R R be the region bounded by y=3x+2,x=4 y = 3x+2 , x = 4, and the x-axis. Let SS be the solid obtained by rotating RR about the axis x=0x=0 . The volume of SS has the form TπT \pi . What is the value of TT?

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