The base of a solid $S$ is the parabolic region $\left\{ ( x , y ) \mid y ^ { 2 } \leq x \leq 1 \right\}$ . Cross-sections perpendicular to the x-axis are squares. Find the volume of $S$ .
A)1.5
B)1.6
C)1.7
D)1.8
E)1.9
F)2.0
G)2.1
H)2.2

Question

Quizplus · Accepted Answer

The volume of the solid $S$ can be calculated by integrating the area of the cross-sections along the x-axis. The area of each square cross-section is $s^2$, where $s$ is the side length of the square. Since the cross-sections are perpendicular to the x-axis and the base is defined by $y^2 \leq x$, for any given $x$, the length of a side of the square is $2y$, because the square spans from $-y$ to $y$ in the y-direction. Therefore, $s = 2y$, and since $y^2 = x$, $s = 2\sqrt{x}$. The area of a cross-section is then $(2\sqrt{x})^2 = 4x$.To find the volume, integrate the area of the cross-sections from the start of the base at $x = 0$ to the end at $x = 1$:$$V = \int_{0}^{1} 4x \, dx = 2x^2 \Big|_{0}^{1} = 2(1)^2 - 2(0)^2 = 2.$$Thus, the volume of the solid $S$ is 2.0.

The Base of a Solid SSS Is the Parabolic Region {(x,y)∣y2≤x≤1}\left\{ ( x , y ) \mid y ^ { 2 } \leq x \leq 1 \right\}{(x,y)∣y2≤x≤1}

The Base of a Solid $S$ Is the Parabolic Region $\left\{ ( x , y ) \mid y ^ { 2 } \leq x \leq 1 \right\}$