Calculus

# Volume of Revolution - Shell Method

The above diagram shows the points $$R=(7,\frac{1}{7})$$ and $$P=(14,\frac{1}{14}),$$ which lie on the curve $$y=\frac{1}{x},$$ point $$Q = (0, \frac{1}{7})$$ on the $$y$$-axis, and points $$A = (7,0)$$ and $$B = (14,0)$$ on the $$x$$-axis. If $$S$$ is the solid obtained by rotating the region $$R$$ bounded by $\overline{OQ}, \overline{QR}, \widehat{RP}, \overline{PB}, \text{ and } \overline{OB}$ about the axis of rotation $$x=-7$$ and the volume of $$S$$ is $$\alpha \pi,$$ what is $$\alpha?$$

Note: The above diagram is not drawn to scale.

Let $$S$$ be the solid obtained by rotating the region $$R$$ bounded by $$y = x^2$$ and $$y = \sqrt{x}$$ about the line $$x=-10$$. If the volume of $$S$$ is $$\alpha \pi,$$ what is $$\alpha?$$

Let $$S$$ be the solid obtained by rotating the region $$R$$ bounded by $$y = x-9, x = 15,$$ and the $$x$$-axis about the axis of rotation $$x=19 .$$ If the volume of $$S$$ is $$\alpha \pi,$$ what is $$\alpha?$$

Let $$S$$ be the solid obtained by rotating the region $$R$$ bounded by $$y = x^3, x = 1, x = 5,$$ and the $$x$$-axis about the line $$x=8$$. If $$10$$ times the volume of $$S$$ is $$\alpha \pi ,$$ what is $$\alpha?$$

Let $$R$$ be the region bounded by $$y = 3x+2 , x = 4$$, and the x-axis. Let $$S$$ be the solid obtained by rotating $$R$$ about the axis $$x=0$$. The volume of $$S$$ has the form $$T \pi$$. What is the value of $$T$$?

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