Sketch the cross-section, (disk, shell, washer) and determine the appropriate formula. Applications of Integrals Brief Review The application of Integrals we will focus on this week is area and volume. Figure 14.2:4 shows the area accumulated from ato x: y-axis orientation is RIGHT-LEFT and your limits come from y. APPLICATION OF INTEGRALS 361 Example 1 Find the area enclosed by the circle x2 + y2 = a2. 4G-6 Find the area of the astroid x2/3 +y2/3 = a2/3 revolved around the x-axis. In all the volume is a a (h2/4)dx = (a 2 − x 2 )dx = 4a 3 /3 −a −a 17. Revolution Applet. With very little change we can find some areas between curves; indeed, the area between a curve and the x-axis may be interpreted as the area between the curve and a second “curve” with equation y = 0. Applications of integration a/2 y = 3x 4B-6 If the hypotenuse of an isoceles right triangle has length h, then its area is h2/4. A … The endpoints of the slice in the xy-plane are y = ± √ a2 − x2, so h = 2 √ a2 − x2. Sketch the area and determine the axis of revolution, (this determines the variable of integration) 2. Sol: We know that the volume of the solid generated by the revolution of the area bounded by the curve , the and the lines is given by Now, given curve Required volume is given by 1. 3. Determine the boundaries of the solid, 4. lated area, length, volume, and surface area, have. AREA BETWEEN TWO CURVES: You can do this with an x-axis orientation in which case your limits of integration come from the x-axis and you do TOP-BOTTOM. APPLICATION OF INTEGRATION Measure of Area Area is a measure of the surface of a two-dimensional region. 1. Determine the boundaries of the solid, 4. 3. 4G-7 Conside the torus of Problem 4C-1. One very useful application of Integration is finding the area and volume of “curved” figures, that we couldn’t typically get without using Calculus. 4. Applications of Integration 9.1 Area between ves cur We have seen how integration can be used to find an area between a curve and the x-axis. Set up the definite integral, and integrate. Sketch the area and determine the axis of revolution, (this determines the variable of integration) 2. Finding volume of a solid of revolution using a disc method. UNIT-4 APPLICATIONS OF INTEGRATION ... Find the volume of the solid that result when the region enclosed by the curve ... Sol: We know that the volume of the solid generated by the revolution of the area bounded by the curve , the and the lines is given by Now, given curve Required volume is given by 3 0 4 CHAPTER 6 APPLICATIONS OF THE DEFINITE INTEGRAL 6.1 AREA FIGURE 6.1 Y a \. APPLICATIONS OF INTEGRATION 4G-5 Find the area of y = x2, 0 ≤ x ≤ 4 revolved around the y-axis. We are familiar with calculating the area of regions that have basic geometrical shapes such as rectangles, squares, triangles, circles and trapezoids. Set up the definite integral, and integrate. Figure 14.2:4 shows the area accumulated from ato x: For example, the accumu-lated area used in the second half of the Fundamental Theorem of Integral Calculus is additive. 1. a) Set up the integral for surface area using integration dx b) Set up the integral for surface area using integration dy In this section we shall consider the lated area, length, volume, and surface area, have. g(.l) h x If a function I is continuous and f(x) 0 on [a, h], then, by Theo- rem (5.19), the area of the region under the graph of f from a to b is given by the definite integral f(x) dx. Title: APPLICATIONS OF INTEGRATION Author: Y.P.REDDY Subject: I YEAR B.Tech Created Date: 4/18/2011 8:01:01 AM Khan Academy Solids of Revolution (10:04) . The left boundary will be x = O and the fight boundary will be x = 4 The upper boundary will be y 2 = 4x The 2-dimensional area of the region would be the integral Area of circle Volume (radius) (ftnction) dx sum of vertical discs') In these two videos, the In the following video the narrator walks trough the steps of setting up a volume integration (14.0)(16.0). For example, the accumu-lated area used in the second half of the Fundamental Theorem of Integral Calculus is additive. Therefore the area of a slice is x2. E. Solutions to 18.01 Exercises 4. Finding volume of a solid of revolution using a disc method. APPLICATIONS OF INTEGRATION I YEAR B.Tech . Sometimes the same volume problem can be solved in two different ways (14.0)(16.0). This means that we can apply Duhamel’s Principle to finding integral formulas of many geometric quantities. This means that we can apply Duhamel’s Principle to finding integral formulas of many geometric quantities. (z/h)M = (z/h)2LM = (z/h)2b Therefore, the volume is h (z/h)2bdz = bz3/3h2 h= bh/3 0 0 4B-4 The slice perpendicular to the xz-plane are right triangles with base of length x and height z = 2x. Problems on Volume of solid of Revolution 1) Find the volume of the solid that result when the region enclosed by the curve is revolved about the . Sketch the cross-section, (disk, shell, washer) and determine the appropriate formula. Volume and Area from Integration a) Since the region is rotated around the x-axis, we'll use 'vertical partitions'. 1.