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Calculus Independent Study Path Practice Unit 8 Test

1. Find the length of the curve

y = \frac{x^{4}}{4} + \frac{1}{8x^{2}}

between x = 1 and x = 2.

2. The arc

x = t + 1, y = \frac{1}{2}t^{2} + t

between t = 0 and t = 4 is revolved about the y-axis. Find the area of the surface produced.

3. The region between x = 0 and x = 2y −y^{2} is revolved about the x-axis. Find the volume produced.

4. Consider the solid formed by revolving the region bounded by y = x^{2} + 1, y = 0, x = 0, and x = 1 about the y-axis. Compute its volume by both the shell and disc methods.

5. Find the length of the curve

x = a cos t + at sin t, y = a sin t − at cos t

between t = 0 and t = π/2; a is a positive constant.

6. The arc

y = \frac{x^{3}}{3} + \frac{1}{4x}

between x = 1 and x = 3 is revolved about the line y = −1. Find the area of the surface produced.