Archimedes<\/span> was fascinated with calculating the areas of various shapes\u2014in other words, the amount of space enclosed by the shape. He used a process that has come to be known as the method of exhaustion<\/em><\/span>, which used smaller and smaller shapes, the areas of which could be calculated exactly, to fill an irregular region and thereby obtain closer and closer approximations to the total area. In this process, an area bounded by curves is filled with rectangles, triangles, and shapes with exact area formulas. These areas are then summed to approximate the area of the curved region.<\/p>\r\n In this section, we develop techniques to approximate the area between a curve, defined by a function [latex]f(x)[\/latex], and the [latex]x[\/latex]-axis on a closed interval [latex][a,b][\/latex]. Like Archimedes, we first approximate the area under the curve using shapes of known area (namely, rectangles). By using smaller and smaller rectangles, we get closer and closer approximations to the area. Taking a limit allows us to calculate the exact area under the curve.\u00a0 Let's first look at some visual examples.<\/p>","rendered":" Archimedes<\/span> was fascinated with calculating the areas of various shapes\u2014in other words, the amount of space enclosed by the shape. He used a process that has come to be known as the method of exhaustion<\/em><\/span>, which used smaller and smaller shapes, the areas of which could be calculated exactly, to fill an irregular region and thereby obtain closer and closer approximations to the total area. In this process, an area bounded by curves is filled with rectangles, triangles, and shapes with exact area formulas. These areas are then summed to approximate the area of the curved region.<\/p>\n In this section, we develop techniques to approximate the area between a curve, defined by a function [latex]f(x)[\/latex], and the [latex]x[\/latex]-axis on a closed interval [latex][a,b][\/latex]. Like Archimedes, we first approximate the area under the curve using shapes of known area (namely, rectangles). By using smaller and smaller rectangles, we get closer and closer approximations to the area. Taking a limit allows us to calculate the exact area under the curve.\u00a0 Let’s first look at some visual examples.<\/p>\n\n\t\t\t What you\u2019ll learn to do:\u00a0Apply summation rules<\/h2>\n
Candela Citations<\/h3>\n\t\t\t\t\t