16.4 Volume of Rotating Solid 旋转体的体积

(1)
Find the volumes generated by rotating the areas of the shaded portion of the following figures about the x-axis.

求下列各图像的阴影部分绕x轴旋转所形成旋转体的体积

Answer :

(a) $\frac{16}{15}\pi$

(b) $\sqrt3\pi$

(c) $15\pi$

(d) $23.4\pi$

(2)
Find the volumes generated by rotating the areas of the shaded portion of the following figures about the y-axis.

求下列各图像的阴影部分绕y轴旋转所形成旋转体的体积

Answer :

(a) $16\pi$

(b) $3\pi$

(3)
Find the volumes generated by rotating the areas of the shaded portion of the following figures about the x-axis and y-axis.

求下列各图像的阴影部分绕x轴和 y轴旋转所形成旋转体的体积

Answer :

(a) $V_x = \frac{64}{15}\pi$ , $V_y= \frac{8}{3}\pi$

(b) $V_x = \frac{4}{3}(\sqrt2 - 1)\pi$ , $V_y = 1\frac{1}{3}\pi$

(4)
Refer to the figure and find the volume generated by rotating the shaded region about the given line

如图，求阴影部分绕下列各直线旋转所形成旋转体的体积

(a) x – axis

(b) x = 1

Answer :

(a) $\frac{32}{15}\pi unit^3$

(b) $\frac{5}{6}\pi unit^3$

(5)
For the region bounded by y = x $^2$ and y = 2 – x $^2$ , calculate
对y = x $^2$ 和 y = 2 – x $^2$ 所包围的区域，计算

(a) the area.
其面积

(b) the volume of the solid generated when the region is rotated through $\pi$ radian about the y-axis.
其对绕y轴旋转所形成旋转体的体积

(c) the volume of the solid generated when the region is rotated through $\pi$ radian about y = 1

Answer :

(a) $2\frac{2}{3} unit^{2}$

(b) $\pi unit^2$

(c) $\frac{16}{15} unit^3$

(6)
The region R, in the positive quadrant, is bounded by the curve y = x $^2$ , the y-axis and the line y = 4. Find the volume of the solid generated when R is rotated through four right angles about

区域R 位于正象限, 被曲线 y = x $^2$ , y-轴和直线 y = 4所包围. 求所形成旋转体的体积当 R绕
(a) y-axis
(b) x-axis
(c) y = 4
(d) x = 2

Answer :

(a) $8\pi$

(b) $25.6\pi$

(c) $\frac{256}{15}\pi$

(d) $13\frac{1}{3}\pi$