3. Volumes

Volume Let’s review how to find volume of a regular geometric prism. Whether it be a circular prism (cylinder), triangular prism, rectangular prism, etc., if we can find the area of the face, we need only to multiply by the distance between the two faces, h. In general V = A*h

Volume The key will be finding the formula for the area of the base V = A * h The key will be finding the formula for the area of the base

Volume with known cross sections Sometimes a shape is “built” by piling shapes with a known cross sections perpendicular to an axis (see projects) If the cross sections are perpendicular to the x-axis, find the area in terms of x If the cross sections are perpendicular to the y-axis, find the area in terms of y

Cross Sections Problems will tell you what cross sections to use. The base of whatever figure it is will be found the same way we set up the integral to find area between 2 curves.

Area formulas for cross sections that show up quite a bit. In each of these cases, b is found as if we were finding the area between curves. http://mathdemos.org/mathdemos/sectionmethod/sectiongallery.html

Example 1 Find the volume of a figure with base between the curves y=x+1 and y=x2-1 with the following cross sections perpendicular to x-axis.

Example 2

Example 3 The base of a solid is bounded by Find the volume of the solid for each of the following cross sections taken perpendicular to the y-axis. (A) Squares  (B) Semicircles   (C) Equilateral triangles

Break!

Rotating a function around an axis Many times a shape is obtained by rotating a function around an axis. This is called a solid of revolution. http://curvebank.calstatela.edu/volrev/volrev.htm Whenever you rotate around an axis perpendicular to the slices, the cross-sections will be circular (DISCs) You can remember this by remembering perperDISCular The area of a circle is πr2. The radius is the value of the function To find volume, take integral of area of cross section

Example 4 Find the volume of the solid formed by rotating the region bounded by the x-axis , and x=1 around the x-axis.

Remember to always draw a picture! Draw a picture of the region and draw the representative slice. This makes the process much easier!

Example 5 Find the volume of the solid formed by rotating the region bounded by y=1, , and x=0 around the line y=1.

Example 6 Find the volume of the solid formed by rotating the region bounded by y=8, , and x=0 around the y-axis.

What if the graph doesn’t reach the axis you are rotating around? Now there will be a void or hole This makes the cross section a washer http://www.mathdemos.org/mathdemos/washermethod/gallery/gallery.html R r Make sure you square each function separately, Don’t subtract first and then square!!

Example 7 Find the volume of the region in the 1st quadrant obtained by rotating the graphs of y = cos x and y = sin x around the x-axis

Important things to consider Always draw a picture! Always draw a picture! Draw R and r on the picture and label them with an equation When writing an equation for R and r, it will still involve TOP – BOTTOM or RIGHT – LEFT. One of these in each case will be the axis of rotation itself. Don’t forget to square each radius before subtracting them! (Did I say that already?)

Example 8 Find the volume of revolving the region enclosed by y = x and y=x2 around

Example 9 Find the volume of the solid formed by rotating the region bounded by , y=0, x=0 and x=1 around the y-axis.

Can we do example 9 with a single integral? Not if we slice perpendicular to the axis of rotation, but what if we slice it more cleverly! What if we sliced the region PARALLEL to the axis of rotation. What would happen to a representative rectangle when taken for a spin around an axis parallel to it? Think “bundt cake”. http://www.mathdemos.org/mathdemos/shellmethod/gallery/gallery.html We call this a Cylindrical Shell.

How do we find the volume of this shell? The shell, with finite thickness has two radii, but as we slice thinner and thinner and thinner, the two radii approach each other. For an infinitely thin slice, we can use a single radius (think school-bought toilet paper) When the volume of solid is obtained by rotating a region paraSHELL to the axis of rotation, the volume is given by

Method Draw a rectangle perpendicular to the axis of revolution – this will be the radius and is what will get revolved around axis Draw a line parallel to axis of revolution inside the area to be revolved – this will be the height and is the function

Example 10 The region enclosed by the x-axis and the parabola is revolved around the line x=-1. What is its volume? What if we rotate around x=4 instead?

Example 11 Find the volume of the solid formed by revolving the region formed by about the y-axis. Use both vertical and horizontal slices. Compare your results.