FINDING VOLUME USING DISK METHOD & WASHER METHOD CHAPTER 4 APPLICATIONS: FINDING AREA FINDING VOLUME USING DISK METHOD & WASHER METHOD

AREA The definite integral represent the area between the function and the x-axis along the interval [a,b].

Area underneath f (x) along [-1,1]

Area underneath g(x) along [-1,1]

Area between f (x) and g(x) along [-1,1] Upper curve Lower curve

Example 1 Find the area bounded by the curve and the x-axis along the interval

Example 2 Find the area bounded by the curve and the x-axis along the interval

Example 3 Find the point of intersection, hence determine the area bounded by the curve and

AREA The definite integral represent the area between the function and the y-axis along the interval [a,b]. Area underneath f (y) along [0,1]

AREA Area underneath g(y) along [0,1]

AREA Area between f(y) and g(y) along [0,1] Right curve Left curve

Example 4 Find the area of the region in graph below:

Example 5 Find the point of intersection, hence determine the area bounded by the curve and

VOLUME ; DISK METHOD Volume of revolution are obtained by revolving a two-dimensional region about an axis of rotation. Given Where Therefore, volume generated on the solid region is given by:

Consider the graph of a function y=f(x) in Quadrant I. For volumes of revolution about the x-axis, the cross-sections of the solid are all circles. Consider the graph of a function y=f(x) in Quadrant I. The region between this graph and the x-axis, between x=1 and x=5, is revolved about the x-axis. x

VOLUME ; DISK METHOD For volumes of revolution about the x-axis: Therefore, volume of revolution about the x-axis is given by:

Example 1 Determine the volume of the solid obtained by rotating the region bounded by and the x-axis about the x-axis. x

Example 2 Determine the volume of the solid obtained by rotating the region bounded by about the x-axis. x

Exercise 2* Find the volume of the solid of revolution formed by rotating the region bounded by the x-axis and the graph of from x=1 to x=2 about the x-axis.

Example 3 Determine the volume of the solid obtained by rotating the region bounded by and the x-axis about the x-axis. x

Consider the graph of a function x=f(y) in Quadrant I. For volumes of revolution about the y-axis, the cross-sections of the solid are all circles. Consider the graph of a function x=f(y) in Quadrant I. The region between this graph and the x-axis, between y=1 and y=4, is revolved about the y-axis.

VOLUME ; DISK METHOD For volumes of revolution about the y-axis: Therefore, volume of revolution about the x-axis is given by:

Example 3 Determine the volume of the solid obtained by rotating the region bounded by and the y-axis about the y-axis. y

Example 4 Determine the volume of the solid obtained by rotating the region bounded by and the y-axis about the y-axis. y

VOLUME ; WASHER METHOD The washer method is essentially but two applications of the disk method.   Suppose that f(x) > g(x) > 0, for all x in the interval I =[a , b]. Further assume that f and g are bounded over that interval. We'll find the volume of the solid of revolution that results when the region bounded by the graphs of f and g over interval I  revolves about the x-axis.

At a point x on the x-axis, a perpendicular cross section of the solid consists of the region between two concentric circles (shaped like a washer, it's called an annulus).   The outer circle has radius  R = f(x), and the inner circle (the hole) has radius    r = g(x). x

VOLUME ; WASHER METHOD For volumes of revolution about the x-axis: Therefore, volume of revolution bounded by two lines about the x-axis is given by:

Example 5 Find the volume of the solid of revolution formed by rotating the finite region bounded by the about the x-axis. x

Example 6 Find the volume of the solid of revolution formed by rotating the finite region bounded by the about the x-axis. x

VOLUME ; WASHER METHOD The washer method is essentially but two applications of the disk method.   Suppose that f(y) > g(y) > 0, for all y in the interval I =[a , b]. Further assume that f and g are bounded over that interval. We'll find the volume of the solid of revolution that results when the region bounded by the graphs of f and g over interval I  revolves about the y-axis.

At a point y on the y-axis, a perpendicular cross section of the solid consists of the region between two concentric circles   The outer circle has radius  R = f(y), and the inner circle (the hole) has radius    r = g(y). x

VOLUME ; WASHER METHOD For volumes of revolution about the y-axis: Therefore, volume of revolution bounded by two lines about the y-axis is given by:

Example 7 Find the volume of the solid of revolution formed by rotating the finite region bounded by the about the y-axis. y

Example 6 Find the volume of the solid of revolution formed by rotating the finite region bounded by the about the y-axis. y