Work
Assume that a spring does not follow Hooke’s Law. Instead, the force required to stretch the spring x ft from its natural length is ƒ(𝓍) = 10𝓍³/² lb . How much work does it take to
a. stretch the spring 4 ft from its natural length?
Find the lengths of the curves in Exercises 19–22.
y = (5/12) x⁶/⁵ ― (5/8)x⁴/⁵ , 1 ≤ x ≤ 32
Volumes
Volume of a solid sphere hole A round hole of radius √3 ft is bored through the center of a solid sphere of radius 2 ft. Find the volume of material removed from the sphere.
Centers of Mass and Centroids
Find the centroid of a thin, flat plate covering the region enclosed by the parabolas 𝔂 = 2𝓍² and 𝔂 = 3 ― 𝓍² .
Volumes
Find the volumes of the solids in Exercises 1–18.
The solid lies between planes perpendicular to the x-axis at x = 0 and x = 1. The cross-sections perpendicular to the x-axis between these planes are circular disks whose diameters run from the parabola y = x² to the parabola y = √x.
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 = √2x + 1 , 0 ≤ x ≤ 3 ; x-axis"
