Finding Cross Section Of Quadratic Function Using Equilateral Triangles

Enter any UPPER FUNCTION labeled upperx. Enter any LOWER FUNCTION labeled lowerx. Enter your lower and upper limits of integration quotaquot and quotbquot, respectively. Slide the FILLING slider all the way to the left. For this solid, note how cross sections parallel to the yAxis are equilateral triangles with one of its sides lying in the base.This side has length upperx - lowerx.

The main problem I am having with this question is setting up the integral, I have not dealt with equilateral triangle and cross sections before so I am having a little trouble. Any help would be appreciated.

How can I find the volume of a solid with a triangular cross section? Use the basic concept. If the area of the cross section of a solid is given by . and is continuous on . Then the volume of the corresponding solid from to is. You may need to create the cross sectional area function based on information provided. For example may depend on the

Using similar triangles, we have the proportion H a H-h b, which gives H ha a-b. bCalculate the side length of a cross-section of the frustum at height x from the base. SOLUTION The cross-section at height x is a square of side length 1hah-xbx. Let w denote the side length of the square cross-section at height x. By similar

based on the given cross sections. Set up the integrals first, then use a calculator to evaluate. 1. Semicircle cross sections perpendicular to the -axis. 0, 2, and 2. Equilateral triangle cross sections perpendicular to the -axis. 3. Isosceles right triangle, with a leg on the base perpendicular to the -axis.

function of y, say Ay, then the volume, V, of the solid on a, b is given by Notice the use of area formulas in order to evaluate the integrals. The area formulas you will need to know in order to do this section include Area of a Square Area of a Triangle Area of an Equilateral Triangle Area of a Circle

Explanation . First, the cross sections being perpendicular to the axis indicates the expression should be in terms of . The area of an equilateral triangle is , with being the side length of the triangle. By applying this formula to our general volume formula , we get the following .The intersection points of the functions and are and .The coordinates of these points will define the

The red strip again represents the box we are using to approximate the water in the 92i92 th subinterval. As noted in the problem statement the cross-section is an equilateral triangle and with sides of length 2 feet. We included the height in the above sketch and this is easy to get using some basic right triangle trig.

1 A description of the base functions used, a sketch of the base region, and an explanation of what each slice looks like. Optional a 3-dimensional sketch of the region using the cross section of your choosing 2 The theoretical volume of the enclosed area with your chosen cross section. Optional Finding the volume

Find V of described triangle. Homework Equations The Attempt at a Solution I first wrote the equation of the line in terms of x x 4-y, which is the base. Since we are dealing with an equilateral triangle the area of the cross section would be Ay 4-y22 and so the integral to calculate the volume is Aydy from 0 to 4.

If the cross sectional area can be expressed as a function of , then the volume is given by where is the -position of the leftmost cross-section and is the -location of the rightmost cross-section. In each case, is the cross-sectional area of a slice of the solid. Note that both the area and the limits of integration must be expressed in terms