## Find the volume of the solid

Math help on Calculating limits of functions, Differentiation of a product, Differentiation of a quotient, Composition of functions, Chain Rule, Variations, Extremums: Maximums, Minimums, Increasing Functions, Decreasing Functions, Constant Functions, Differential Equations, Growth of a Population, Formulas, Integration by Parts, Change of Variables; Double Integrals, Triple Integrals, Multiple Integrals, Integrals with a Parameter, Area, Volume, Approximations on My Math Forum.

### Find the volume of the solid

Find the volume of the solid s

The base of S is an elliptical region with boundary curve 9x^2 + 4y^2 = 36. Cross-sections perpendicular to the x-axis are isosceles right triangles with hypotenuse in the base.

amiv4
Newcomer

Posts: 1
Joined: Thu Oct 02, 2008 11:03 pm

### Re: Find the volume of the solid

Unless I'm wrong, there is not sufficient information. I'm thinking it's an elliptical cone with the apex on the z-axis ...but it doesn't state how far along in that direction. You can't find volume if all you have is the base.
Dave

Posts: 895
Joined: Mon Jul 28, 2008 9:21 pm

### Re: Find the volume of the solid

Yes, Dave, we can find the volume. This is not a volume of revolution.

Here is a similar problem. Use this to see if you can figure out yours...OK.

"The base of a solid is the region enclosed by $y=\frac{1}{x}, \;\ y=0, \;\ x=1, \;\ x=3$.

Every cross section perp. to the x-axis is an isosceles triangle with hypoteneuse across the base. Find the volume."

SOLUTION:

Use the formula for the area of the triangle and sub it into the integral.

We get $\frac{1}{2}\left(\frac{1}{2}\cdot\frac{1}{x}\right)\left(\frac{1}{x}\right)=\frac{1}{4x^{2}}$

$V=\int_{1}^{3}\frac{1}{4x^{2}}dx=\frac{1}{6}$

Can you try yours now?. You have an ellipse with minor axis of length 2 along the x-axis.

A lot of times we see these problems with semicircles as cross sections, squares, equilateral triangles, and so forth.

For a semicircle we would use the area of a semicircle, etc.
galactus