# Solids of Known Cross sections(volume)

• Mar 19th 2009, 08:37 PM
dandaman
Solids of Known Cross sections(volume)
The base of a solid is bounded by y=x^3, y=0, and x=1. Find the volume of the solid for each of the following cross sections perpendicular to the y-axis.

a)semicircles

b)Squares

-MY STEPS

a) $A =\frac{\pi r^2}{2}$

$r = (\sqrt[3]{y/2})$

$V = \pi /4 \int_{0}^{1} (\sqrt[3]{y})^2$

V=.4712319
(Did I do this right?)

b) $A=D^2$

$D=\sqrt[3]{y}$
$V=\int_{0}^{1}(\sqrt[3]{y})^2$
V=.600000
(did I do this right?)

I don't think I did these right. Can someone help me out. Thanks.
• Mar 20th 2009, 05:02 AM
Volume problems
Hello dandaman
Quote:

Originally Posted by dandaman
The base of a solid is bounded by y=x^3, y=0, and x=1. Find the volume of the solid for each of the following cross sections perpendicular to the y-axis.

a)semicircles

b)Squares

-MY STEPS

a) $A =\frac{\pi r^2}{2}$

$r = (\sqrt[3]{y/2})$

$V = \pi /4 \int_{0}^{1} (\sqrt[3]{y})^2$

V=.4712319
(Did I do this right?)

b) $A=D^2$

$D=\sqrt[3]{y}$
$V=\int_{0}^{1}(\sqrt[3]{y})^2$
V=.600000
(did I do this right?)

I don't think I did these right. Can someone help me out. Thanks.

$y = x^3 \Rightarrow \frac{dy}{dx} = 3x^2 \Rightarrow dy = 3x^2dx$

(a) Take a cross-section at right-angles to the y-axis at the point $(x, y)$ on $y = x^3$. Then the semi-circle has diameter $(1-x)$, and area $\pi(\tfrac{1}{2}(1-x))^2 = \frac{\pi}{4}(1-x)^2$. So the volume is:

$\int_0^1\frac{\pi}{4}(1-x)^2dy=\int_0^1\frac{\pi}{4}(1-x)^23x^2\,dx$

(b) With a square cross-section, the area of cross-section is $(1-x)^2$. So the volume is

$\int_0^1(1-x)^23x^2\,dx$

Can you complete them now?