Question

A solid lies between planes perpendicular to the​ x-axis at x=0 and x=8. The​ cross-sections perpendicular to the axis on the interval 0

Answer 1

Answer:

The volume of the solid is 256 cubic units.

Step-by-step explanation:

Given:

The solid lies between planes $x=0\ and\ x=8$

The cross section of the solid is a square with diagonal length equal to the distance between the parabolas $y=-2\sqrt{x}\ and\ y=2\sqrt{x}$.

The distance between the parabolas is given as:

$D=2\sqrt x-(-2\sqrt x)\\\\D=2\sqrt x+2\sqrt x\\\\D=4\sqrt x$

Now, we know that, area of a square with diagonal 'D' is given as:

$A=\frac{D^2}{2}$

Plug in $D=4\sqrt x$. This gives,

$A=\frac{(4\sqrt x)^2}{2}\\\\A=\frac{16x}{2}\\\\A=8x$

Now, volume of the solid is equal to the product of area of cross section and length $dx$. So, we integrate it over the length from $x=0\ to\ x=8$. This gives,

$V=\int\limits^8_0 {A} \, dx\\\\V=\int\limits^8_0 {(8x)} \, dx\\\\V=8\int\limits^8_0 {(x)} \, dx\\\\V=8(\frac{x^2}{2})_{0}^{8}\\\\V=4[8^2-0]\\\\V=4\times 64\\\\V=256\ units^3$

Therefore, the volume of the solid is 256 cubic units.