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Volume. Find the volume of the solid formed by revolving the region bounded by the graphs y = x 3 + x + 1, y = 1, and x = 1 about the line x = 2.

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Presentation on theme: "Volume. Find the volume of the solid formed by revolving the region bounded by the graphs y = x 3 + x + 1, y = 1, and x = 1 about the line x = 2."— Presentation transcript:

1 volume

2 Find the volume of the solid formed by revolving the region bounded by the graphs y = x 3 + x + 1, y = 1, and x = 1 about the line x = 2.

3 The region between the curve, and the y -axis is revolved about the y -axis. Find the volume. y x We use a horizontal disk. The thickness is dy. The radius is the x value of the function. volume of disk

4

5 x y A 45 o wedge is cut from a cylinder of radius 3 as shown. Find the volume of the wedge. If we let h equal the height of the slice then the volume of the slice is: Since the wedge is cut at a 45 o angle: x h 45 o Since

6 x y Even though we started with a cylinder,  does not enter the calculation!

7 Washer Cross Section 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.

8 Find the volume of the soild formed by revoling the region bouned by the graph of : about the x-axis

9 Find the volume of the soild formed by revoling the region bouned by the graph of :

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11

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