Let $R$ be the region in the first quadrant bounded below by the graph of $y = x ^ { 2 }$ and above by the graph of $y = \sqrt { x }$. $R$ is the base of a solid whose cross sections perpendicular to the $x$-axis are squares. What is the volume of the solid?
(A) 0.129
(B) 0.300
(C) 0.333
(D) 0.700
(E) 1.271

Let $R$ be the region in the first quadrant bounded above by the graph of $y = \ln ( 3 - x )$, for $0 \leq x \leq 2$. $R$ is the base of a solid for which each cross section perpendicular to the $x$-axis is a square. What is the volume of the solid?
(A) 0.442
(B) 1.029
(C) 1.296
(D) 3.233
(E) 4.071

As shown in the figure, there is a solid figure with base formed by the curve $y = \sqrt { x } + 1$, the $x$-axis, the $y$-axis, and the line $x = 1$. When the cross-section of this solid figure cut by a plane perpendicular to the $x$-axis is always a square, what is the volume of this solid figure? [3 points]
(1) $\frac { 7 } { 3 }$
(2) $\frac { 5 } { 2 }$
(3) $\frac { 8 } { 3 }$
(4) $\frac { 17 } { 6 }$
(5) 3

As shown in the figure, for a positive number $k$, the region enclosed by the curve $y = \sqrt { \frac { e ^ { x } } { e ^ { x } + 1 } }$, the $x$-axis, the $y$-axis, and the line $x = k$ is the base of a solid figure. When the cross-section perpendicular to the $x$-axis is always a square and the volume is $\ln 7$, what is the value of $k$? [3 points]
(1) $\ln 11$
(2) $\ln 13$
(3) $\ln 15$
(4) $\ln 17$
(5) $\ln 19$

As shown in the figure, there is a solid figure with base formed by the curve $y = \sqrt{(1-2x)\cos x}$ ($\frac{3}{4}\pi \leq x \leq \frac{5}{4}\pi$) and the $x$-axis and the two lines $x = \frac{3}{4}\pi$ and $x = \frac{5}{4}\pi$. When this solid figure is cut by a plane perpendicular to the $x$-axis, all cross-sections are squares. Find the volume of this solid figure. [3 points]
(1) $\sqrt{2}\pi - \sqrt{2}$
(2) $\sqrt{2}\pi - 1$
(3) $2\sqrt{2}\pi - \sqrt{2}$
(4) $2\sqrt{2}\pi - 1$
(5) $2\sqrt{2}\pi$