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Area, Volume, and Arc Length

Area, Volume, and Arc Length

Example 1

Let R be the region bounded by y = ln x and the x-axis on the interval [1,2]. Find an integral expression for the volume of the solid obtained by rotating R around

(a) the x-axis

(b) the line x = 2

Example 2

Let R be the region bounded by y = x3, the y-axis, and the line y = 8. Find an integral expression for the volume of the solid obtained by rotating R around

(a) the y-axis

(b) the line y = 8

Example 3

Let R be the region bounded by the graphs y = 1 – x, y = x2 + 1, and the line x = 1.

Find an integral expression for the volume of the solid obtained by rotating R around the line

(a) y = 0

(b) y = 2

(c) y = -2

(d) y = 5

Example 4

Let R be the portion of the unit circle that falls in the first quadrant. Find an integral expression for the volume of the solid obtained by rotating R around the line

(a) y = 0

(b) y = 2

(c) x = 1

(d) x = -1

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