Math 2414 Section 6 2 Regions between Curves Finding Area by Integration Recall from Section 5 3 the area under the graph of a nonnegative function f over the interval b a b is the definite integral f x dx a Find the area of the region bounded by the graph of and x 3p 4 f x sin 2 x and the x axis between x 0 y x Also recall that the properties of definite integrals see Section 5 2 allow us to calculate areas of regions separately Find the area of the region bounded by the graph of f x tan x and the x axis on p 3 0 y x Setup the integrals needed to find the area of the region bounded above by the graph of y x 3 and below by the x axis for 1 x 6 y x The figure below shows a region bounded by the graphs of two continuous functions above by y f x y g x the graph of and below by the graph of for values of x between a and b We can use the definite integral to compute the area of such regions First we divide the interval b a D x a b into n subintervals of equal width n On each subinterval we build a rectangle th extending from the lower curve to the upper curve On the k subinterval a point xk is chosen f xk g xk and the height of the corresponding rectangle is taken to be Therefore the area th f xk g xk Dx of the k rectangle is Summing the areas of the n rectangles gives an approximation to the area of the region between the curves The limit of this Riemann sum is a definite integral of the function f g f x g x is the length of a rectangle and dx is its f x g x dx width We sum integrate the areas of the rectangles to obtain the area of the region It is helpful to interpret the area formula as Definition Area of a Region between Two Curves f x g x a b Suppose that f and g are continuous functions with on the interval The a b area of the region bounded by the graphs of f and g on is b A f x g x dx a 1 1 f x g x x 2 1 x 2 between the Find the area of the region bounded by the graphs of y axis and x 1 y x Setup the integrals needed to find the area of the region between the graphs of 1 g x 2 for 0 x 2p f x sin x and y x Integrating with Respect to y There are occasions when it is convenient to reverse the roles of x and y For example 2 consider the area bounded by the graphs of the equations y x and x y 2 y y x x By using vertical rectangles one encounters a difficulty some approximating rectangles have 2 both bottoms and tops on the graph of x y 2 while others have tops on the graphs of x y 2 2 and bottoms on the graph of y x A simpler approach is to partition the y axis 2 Then all approximating rectangles run from the curve x y 2 to the line x y 2 Find the area bounded by the graphs of the equations y x and x y 2 Definition Area of a Region between Two Curves with Respect to y f y g y c d Suppose that f and g are continuous functions with on the interval The x f y x g y c d area of the region bounded by the graphs and on is d A f y g y dy c For the following areas should the variable of integration be x or y y y x x y y x x y y x x Setup the integrals needed to find the area of the region bounded by the graphs of the equations 3 y x 6 x y 2 and x y 2 4 as illustrated below y x