nick's question at Yahoo! Answers regarding a volume by slicing

MarkFL

Staff member
Here is the question:

Volume of solid (calc 2)?

Find the volume V of the solid whose base is the circle
x^2 + y^2 = 25
and whose cross sections perpendicular to the x-axis are triangles whose height and base are equal.

help appreciated

thanks
I have posted a link there to this thread so the OP can view my work.

MarkFL

Staff member
Hello nick,

For an arbitrary slice of the described solid, the base of this triangular slice will be from the $y$-coordinate of the upper half to the $y$-coordinate of the lower half, or:

$$\displaystyle b=y-(-y)=2y$$

And thus, since the base and height are the same, and using the formula for the area of a triangle, we find the volume of the slice is:

$$\displaystyle dV=\frac{1}{2}(2y)(2y)\,dx=2y^2\,dx$$

Now, using the boundary of the base of the solid, we find:

$$\displaystyle 2y^2=2\left(25-x^2 \right)$$

And so we obtain:

$$\displaystyle dV=2\left(25-x^2 \right)\,dx$$

Now, summing up the slices, we get:

$$\displaystyle V=2\int_{-5}^{5}25-x^2\,dx$$

And using the even-function rule, we may write:

$$\displaystyle V=4\int_{0}^{5}25-x^2\,dx$$

Applying the FTOC, there results:

$$\displaystyle V=4\left[25x-\frac{1}{3}x^3 \right]_{0}^{5}=4\cdot5^3\left(1-\frac{1}{3} \right)=\frac{(2\cdot5)^3}{3}=\frac{1000}{3}$$