Review: Area betweens two curves Find the area of the region between the curves 𝑓 𝑥 = 𝑥 , 𝑔 𝑥 =𝑥−2 𝑎𝑛𝑑 𝑦=0

Sec 6.2 Volumes Definition: Volumes of Solids with Known Cross Section The cross-sectional area A(x) will vary as x increases from a to b.

Example 1 Show that the volume of a sphere of radius r is .

Example 2 Find the volume of the solid generated. is rotated about the x-axis on [0, 4]. Find the volume of the solid generated.

Example 3 Find the volume of the solid generated by revolving a region between the y-axis and the curve x = 2/y from y = 1 to y = 4

Washers If the region revolved does not border on or cross the axis of revolution, the solid has a hole in it. The cross sections perpendicular to the axis are washers. V = Outside Volume – Inside Volume

Example 4 The region bounded by the curve y = x2 +1 and the line y = -x + 3 is revolved about the x-axis to generate a solid. Find the volume of the solid of revolution.

Example 5 The region bounded by the parabola y = x2 and the line y = 2x in the first quadrant is revolved about the y-axis to generate a solid. Find the volume of the solid.

Example 6 Find the volume of a pyramid whose base is a squre with side L and height is h.