# Thread: Solids of Known Cross sections(volume)

1. ## 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)$\displaystyle A =\frac{\pi r^2}{2}$

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

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

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

b)$\displaystyle A=D^2$

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

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

2. ## Volume problems

Hello dandaman
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)$\displaystyle A =\frac{\pi r^2}{2}$

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

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

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

b)$\displaystyle A=D^2$

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

I don't think I did these right. Can someone help me out. Thanks.
$\displaystyle y = x^3 \Rightarrow \frac{dy}{dx} = 3x^2 \Rightarrow dy = 3x^2dx$

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

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

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

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

Can you complete them now?