Volumes
Find the volume of the solid generated by revolving the region bounded on the left by the parabola x = y² + 1 and on the right by the line x = 5 about
a. the x-axis

Areas of Surfaces of Revolution

In Exercises 23–26, find the areas of the surfaces generated by revolving the curves about the given axes.

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y = √4y ― y² , 1 ≤ y ≤ 2 ; y-axis

Volumes

Find the volume of the solid generated by revolving the region bounded by the curve y = sin x and the lines x = 0, x = π and y = 2 about the line y = 2.

Volumes

Find the volume of the solid generated by revolving the region bounded by the x-axis, the curve y = 3x⁴ , and the lines x = 1 and x = ―1 about

c. the line x = 1

Volumes

Find the volumes of the solids in Exercises 1–18.

The solid lies between planes perpendicular to the x-axis at x = π/4 and x = 5π/4. The cross-sections between these planes are circular disks whose diameters run from the curve y = 2 cos x to the curve y = 2 sin x.

Volumes

Find the volume of the solid generated by revolving the region between the x-axis and curve y = x² ―2x about

b. the line y = ―1

Find the lengths of the curves in Exercises 19–22.

y = (5/12) x⁶/⁵ ― (5/8)x⁴/⁵ , 1 ≤ x ≤ 32