Volumes of Revolution

Volumes of Revolution We’ll first look at the area between the lines y = x , . . . x = 1, . . . and the x-axis. 1 Can you see what shape you will get if you rotate the area through about the x-axis? Ans: A cone ( lying on its side )

Volumes of Revolution We’ll first look at the area between the lines y = x , . . . x = 1, . . . r and the x-axis. h 1 For this cone,

Volumes of Revolution The formula for the volume found by rotating any area about the x-axis is x a b Where is the curve forming the upper edge of the area being rotated. a and b are the x-coordinates at the left- and right-hand edges of the area. We leave the answers in terms of

r h So, for our cone, using integration, we get Volumes of Revolution So, for our cone, using integration, we get We must substitute for y using before we integrate. r h 1 Now we’ll outline the proof of the formula ...

The formula can be proved by splitting the area into narrow strips Volumes of Revolution The formula can be proved by splitting the area into narrow strips . . . x which are rotated about the x-axis. Each tiny piece is approximately a cylinder ( think of a penny on its side ). Each piece, or element, has a volume

The formula can be proved by splitting the area into narrow strips Volumes of Revolution The formula can be proved by splitting the area into narrow strips . . . x which are rotated about the x-axis. Each tiny piece is approximately a cylinder ( think of a penny on its side ). Each piece, or element, has a volume

The formula can be proved by splitting the area into narrow strips Volumes of Revolution The formula can be proved by splitting the area into narrow strips . . . x which are rotated about the x-axis. Each tiny piece is approximately a cylinder ( think of a penny on its side ). Each piece, or element, has a volume The formula comes from adding an infinite number of these elements.

through radians about the x-axis. Find the volume of the solid formed. Volumes of Revolution e.g. 1(a) The area formed by the curve and the x-axis from x = 0 to x = 1 is rotated through radians about the x-axis. Find the volume of the solid formed. (b) The area formed by the curve , the x-axis and the lines x = 0 and x = 2 is rotated through radians about the x-axis. Find the volume of the solid formed. Solution: To find a volume we don’t need a sketch unless we are not sure what limits of integration we need. However, a sketch is often helpful. As these are the first examples we’ll sketch the curves.

(a) rotate the area between Volumes of Revolution (a) rotate the area between rotate about the x-axis area A common error in finding a volume is to get wrong. So beware!

Volumes of Revolution (a) rotate the area between a = 0, b = 1

Volumes of Revolution

(b) Rotate the area between and the lines x = 0 and x = 2. Volumes of Revolution (b) Rotate the area between and the lines x = 0 and x = 2.

(b) Rotate the area between and the lines x = 0 and x = 2. Volumes of Revolution (b) Rotate the area between and the lines x = 0 and x = 2.

Volumes of Revolution Remember that

radians about the x-axis. Find the volume of the solid formed. Volumes of Revolution Exercise radians about the x-axis. Find the volume of the solid formed. 1(a) The area formed by the curve the x-axis and the lines x = 1 to x = 2 is rotated through (b) The area formed by the curve , the x-axis and the lines x = 0 and x = 2 is rotated through radians about the x-axis. Find the volume of the solid formed.

, the x-axis and the lines x = 1 and x = 2. Volumes of Revolution Solutions: 1. (a) , the x-axis and the lines x = 1 and x = 2.

Volumes of Revolution Solutions:

, the x-axis and the lines x = 0 and x = 2. Volumes of Revolution (b) , the x-axis and the lines x = 0 and x = 2. Solution:

Rotation about the y-axis Volumes of Revolution Rotation about the y-axis To rotate an area about the y-axis we use the same formula but with x and y swapped. Tip: dx for rotating about the x-axis; dy for rotating about the y-axis. The limits of integration are now values of y giving the top and bottom of the area that is rotated. As we have to substitute for x from the equation of the curve we will have to rearrange the equation.

Volumes of Revolution e.g. The area bounded by the curve , the y-axis and the line y = 2 is rotated through about the y-axis. Find the volume of the solid formed.

Volumes of Revolution

the y-axis and the line y = 3 is rotated through Volumes of Revolution Exercise the y-axis and the line y = 3 is rotated through radians about the y-axis. Find the volume of the solid formed. 1(a) The area formed by the curve for (b) The area formed by the curve , the y-axis and the lines y = 1 and y = 2 is rotated through radians about the y-axis. Find the volume of the solid formed.

For , the y-axis and the line y = 3. Volumes of Revolution Solutions: (a) For , the y-axis and the line y = 3.

, the y-axis and the lines y = 1 and y = 2. Volumes of Revolution (b) , the y-axis and the lines y = 1 and y = 2. Solution: