Volumes By Cross Sections

A squares b equilateral triangles c semicircles d isosceles triangles with the hypotenuse as the base of the solid 3.

Volumes by cross sections. To that end we first consider another familiar shape in preview activity pageindex 2. And in fact if you re not i encourage you to review that on khan academy. We consider solids whose cross sections are common shapes such as triangles squares rectangles trapezoids and semicircles.

Video transcript instructor you are likely already familiar with finding the area between curves. If the cross sections are perpendicular to the y axis then their areas will be functions of y denoted by a y in this case the volume v of the solid on a b is example 1. But what we re going to do in this video.

Let s be a solid and suppose that the area of the cross section in the plane perpendicular to the x axis is a left x right for a le x le b figure 1. Because the cross sections are squares perpendicular to the y. If the cross section is perpendicular to the x axis and itʼs area is a function of x say a x then the volume v of the solid on a b is given by.

Squares and rectangles intro volume with cross sections. Volumes with known cross sections if we know the formula for the area of a cross section we can find the volume of the solid having this cross section with the help of the definite integral. Volumes with cross sections.

For example we could find this yellow area using a definite integral. This is the currently selected item. Indicated cross sections taken perpendicular to the x axis.

Volume of a solid using integration. The principal problem of interest in our upcoming work will be to find the volume of certain solids whose cross sections are all thin cylinders or washers and to do so by using a definite integral. Find the volume of the solid for each of the following cross sections taken perpendicular to the y axis.