Thread: Volume by Slicing - # 2

1. Volume by Slicing - # 2

The solid lies between planes perpendicular to the x axis at $x = -1$ and $x = 1$. The cross-sections perpendicular to the x axis between these planes are squares whose bases run from the semi-circle $y = -\sqrt{1 - x^{2}}$ to the semi-circle $y = \sqrt{1 - x^{2}}$
$V = \int_{a}^{b} A(x) dx$

$x^{2}$ is the area of a square

$A = x^{2}$

$V = \int_{-1}^{1} [2 \sqrt{1 - x^{2}}]^{2}$ On the right track here?

$V = \int_{-1}^{1} [4 (1 - x^{2})]$

$V = \int_{-1}^{1} 4 - 4x^{2}$

The book says the answer will come out to $\dfrac{16}{3}$

2. Re: Volume by Slicing - # 2

Are you supposed to rotate this region about an axis? As it is, you have described a 2 dimensional region, so there is no volume...

- Hollywood

4. Re: Volume by Slicing - # 2

The solid lies between planes perpendicular to the x axis at $x = -1$ and $x = 1$. The cross-sections perpendicular to the x axis between these planes are squares whose bases run from the semi-circle $y = -\sqrt{1 - x^{2}}$ to the semi-circle $y = \sqrt{1 - x^{2}}$
$V = \int_{a}^{b} A(x) dx$

$x^{2}$ is the area of a square

$A = x^{2}$

$V = \int_{-1}^{1} [x][2 \sqrt{1 - x^{2}}]^{2} dx$

$V = \int_{-1}^{1} [x][4 (1 - x^{2})] dx$

$V = \int_{-1}^{1} [x][4 - 4x^{2})] dx$

$V = \int_{-1}^{1} 4x - 4x^{2})] dx$

$V = \dfrac{4x^{2}}{2}- \dfrac{4x^{3}}{3})]$

$V = 2x^{2} - \dfrac{4x^{3}}{3})]$