Area Bounded By Curves

Transcription

1 Area under Curves The most important topic of Integral calculus is Calculation of area. Integration in general is considered to be a tough topic and area calculation tests a person s Integration and that too definite integral which is all the more difficult. Integration including both Definite and Indefinite integrals lays the groundwork for the questions of area calculation in Integral calculus. In this chapter we have basically discussed the basic concepts of area under curves followed by the working rule of finding the area. The concepts have been explained in detail along with supportive illustrations wherever required. We have also included the concept of definite integration and its application for calculation of area of the regions bounded by specific curves. We have tried to cover all the major types of questions which are likely to be encountered by students in various competitive exams. Area Calculation is important from the perspective of scoring high in IIT JEE as there are few fixed patterns on which a number of Multiple Choice Questions are framed. You are expected to do all the questions based on this to remain competitive in IIT JEE examination. We have learnt that the definite integral between two values of Independent variable represents the area of the curve bound by the curve, the axis of the independent variable. Further, as we can calculate the area under one curve and the area under another curve then we can calculate the area between two curves. Depending upon the nature of the curves, this area can have different shapes and thus the tool of definite integral can be employed to calculate the area of different shapes. As a matter of fact, you will realize that the standard formulae to calculate the areas of different shapes can be derived by definite integral by choosing the appropriate curves. Various sub heads included under this topic are listed below: Area under Curves Basic Concepts The calculation of area under curves is a very important topic under integral calculus. In such questions it is indispensable to solve a question without the graph. So once you draw a graph, it gives a clear picture of the question and the limits of integration too. Hence, students are advised to focus on their curve tracing skills so as to solve the questions of area under curves easily and quickly. The working rules to compute area have been discussed in the coming sections. We shall discuss here some of the basic concepts of area under curves:

2 The procedure to calculate the area under a curve f(x) is given by the formula A = The geometrical interpretation of a definite integral is that it gives the area included between the curve y = f(x), the x-axis and the ordinates x = a and x = b. Consider the curve shown in the above figure. We take a strip of width δx and length y from x-axis. Clearly Area EFGJ < Area FEIG < Area FEIK. This means yδx < Area FEIG < (y + δy).δx Also, as δx 0 the area of lower and upper rectangle tend to be equal. Thus, by sandwich theorem Area FEIG = ydx or dδ = ydx. Hence, the area of the region ABCDA is given by Some Important Facts: 1. Area bounded by the curve y 2 = 4ax and x 2 = 4by is 16ab/3. This result can be easily proved by computing the points of intersection of both and then applying the above formula. 2. Area bounded by the parabola y 2 = 4ax and y = mx is equal to 8a 2 /3m 3.

3 3. Area enclosed between the curve y 2 = 4ax and its double ordinate at x = a is 8a 2 /3: 4. The area of the whole ellipse is equal to πab. 5. Let f(x) be a continuous function in (a, b). Then the area bounded by the curve y = f(x), the x-axis and the lines x = a and x = b is given by the formulae provided f(x) > 0 (or f(x) < 0)) x (a, b).

4 6. The area bounded by x = f(y), the y-axis and the lines y = c and y = d is given by provided f(y) > 0 (or f(y) < 0)) y (a, b). 7. The area enclosed by a curve remains invariant if the coordinate axis are shifted and hence shifting of origin can be utilized in attempting such questions as it simplifies the computation of area. 8. In order to draw the curve, some standard steps to be followed are: (a) Check the symmetry: (i) Symmetry about x- axis: If when y is changed to (-y), the curve remains unchanged, then the curve is said to be symmetrical about the x-axis. (ii) Symmetry about y- axis: If when x is changed to (-x), the curve remains unchanged, then the curve is said to be symmetrical about the x-axis. (iii) Symmetry about both the axis: If the curve remains unchanged when x is changed to (-x) and when y is changed to (-y) then the curve is said to be symmetrical about both the axis.

5 (iv) Symmetry about the line y = x: If the curve remains unchanged on interchanging x and y, then the curve is said to be symmetrical about the line y = x. (b) Figure out the points where the curve intersects the x and the y-axis. (c) Compute dy/dx to identify the stationary points. If possible, try to search for the intervals where f(x) is increasing or decreasing. (d) Try to identify y when x or -. The above cases generally cover most of the questions of area. It is advised to shade the area in the graph as done above so that it becomes simple to get the limits and the chances of committing errors are also minimized. Working Rule to Calculate Area Area under curves is an extremely important topic of mathematics syllabus of IIT JEE. Questions can be asked on computation of area of any kind of curve and hence students are advised to have mastery in curve tracing. If a person is able to draw a curve correctly, then it becomes easy to reach at the accurate solution. In this section, we shall discuss some of the ways of calculating area in case of standard curves followed by working rules of calculating area. (i) To determine the area enclosed between the curves (a) y = f(x), x = a and x = b, (b) x = f(y), y = a and y = b. Case (a) The area bounded by the curve y = f(x), x-axis and the ordinates x = a and x = b (where b > a) is given by the formula

6 Case (b) The area bounded by the curve x = f(y), y-axis and the ordinates y = a and y = b (where b > a) is given by the formula (ii) To find the area bounded by the straight lines x = a, x = b (where a < b) and the curves y = f(x) and y = g(x), where f(x) < g(x) is given by Similarly, the area bounded by the straight lines y = a, y = b (where a < b) and the curves x = f(y) and x = g(y), where f(y) < g(y) is given by (iii) To find the area bounded by the curves y = φ(x) and y = Ψ(x) If the roots of the intersection are x = x 1 and x = x 2, then the required area is given by where x 1 < x 2 and x 1 and x 2 are the roots of the equation f(x) = g(x). Similarly, if two curves x = f(y) and x = g(y) intersect, the required bounded area is given by

7 here x 1 < x 2 and x 1 and x 2 are the roots of the equation f(y) = g(y). (iv) Area between the curve y = f(x) and y-axis: To obtain the area between the curve and the y-axis, the function must be written in y. (see figure A and B) i.e. y = f(x) must be inverted to x = g(y) (where g(x) = f -1(x)) and the integral to be evaluated is A= Similarly the area bounded between the y-axis & the curves y = f(x) and y = g(x) can be determined. In general, to find the area of the region one must draw the curve and locate the region. The limits and sign of different definite integral are determined accordingly. (v) If the curve lies completely above the x-axis, then the area is positive but when it lies completely below the x-axis then the area is negative; however, we have the convention to consider the magnitude only. (vi) If the curve lies on both the sides of the x-axis i.e. above the x-axis as well as below the x-axis, then calculate both areas separately and add their moduli to get the total area.

8 In general if the curve y = f(x) crosses the x-axis n times when x varies from a to b, then the areas between y = f(x), the x-axis and the lines x = a and x = b is given by A = A 1 + A A n. If the curve is symmetrical about the x-axis, or the y-axis, or both, then calculate the area of one symmetrical part and multiply it by the number of symmetrical parts to get the whole area. Area Enclosed Between the Curves Area between curves y = Φ(x) and y = Ψ(x) and ordinates x = x1 and x = x2 (i) To determine the area between curves, first find out the points of intersection of the two curves. (See figure) (ii) If in the domain common to both (i.e. the domain given by the points of intersection) the curves lie above x-axis, then area is Note: If however one part of one or both the curve lies below x-axis, then the individual integral must be evaluated according to the case considered in the last topic.

9 3. Area between curve y = f(x) and y-axis. To obtain the area between the curve and the y-axis, the function must be written in y. (see figure A and B) i.e. y = f(x) must be inverted to x = g(y) (where g(x) = f 1 (x)) and the integral to be evaluated is y1 y2 x dy or y1 y2 g(y) dy. Similarly the area bounded between the y-axis & the curves y = f(x) and y = g(x) can be determined. In general, to find the area of the region one must draw the curve and locate the region. The limits and sign of different definite integral are determined accordingly. Generally speaking, when we aim at calculating the area bounded by a curve, we have a figure of the type given below: Here, the area S covered by the curve f(x) is the area we wish to calculate. The area S covered by the curve f(x) between x = a and x = b can be found by integrating the curve f(x) from a to b. Some of the basic points to be kept in mind while dealing with the questions of this topic are: A graph is of utmost importance in these questions. The bounding region provides the limits of integration and it is not easy to do that without a graph.

10 It is very confusing to determine which function is an upper function and which is lower without a graph. So in order to avoid any mistake, students are advised to first draw a graph to the question so as to have a clear picture of what exactly is being asked. The area between the graph y= f(x) and the x-axis is given by the definite integral as given below. This formula gives a positive answer for a graph above the x-axis and a negative answer for the one below the x- axis. In case, the graph is partly below and partly above the x-axis, the formula gives the net resultant area i.e. the area above minus the area below the x-axis. Before we try attempting questions on area under curves, it is important to have an idea about the concepts related to curves. We first throw some light on such concepts: Conditions for various asymptotes: 1. If lim x a f(x) = or lim x a f(x) = -, then x = a is an asymptote of y = f(x). 2. If lim x + f(x) = k or lim x - f(x) = k, then y = k is an asymptote of y = f(x). 3. If lim x - f(x)/x = a 2 and lim x - (f(x) a 2x) = c*, then y = a 2x + c* is an asymptote which is inclined towards the left. 4. If lim x f(x)/x = a 1 and lim x (f(x) a 1x) = c, then y = a 1x + c is an asymptote which is inclined towards the right. A student needs to be really proficient in curve tracing as if one is able to draw the curve of the function, then it becomes quite simple to find the required area. We discuss here some of the ways which can help a student in judging the type of curve: 1. If all the powers of y in the equation of curve are even, then the curve is symmetrical about the x- axis. For eg: y 2 = 4ax

11 2. Similarly, if all the powers of y in the equation of curve are even, then the curve is symmetrical about the y-axis. For eg: x 2 = 4ay 3. If all the powers of x and y in the equation of the curve are even, then the curve is symmetrical both about x as well as y-axis. Eg: x 2 + y 2 = a 2 4. If the equation of the curve remains unchanged on interchanging x and y, then the curve is symmetrical about the line y = x.

12 5. If the equation of the curve remains unaltered when x and y are replaced by their negatives i.e. x and y respectively, the curve is symmetric in opposite quadrants. Eg: xy = c 2 Various working rules for calculation of area have been discussed in the coming sections. Thus, students wanting to get an in-depth knowledge of area under curves can refer the later sections. Some Key Facts: The curve which lies completely above the x-axis is obviously positive while if it lies below the x- axis then the area is negative. By convention, only the magnitude of the area is taken into consideration. If the curve lies completely above or below the x-axis then it can be easily decided whether it will be positive or negative. But, at times, the curve may lie partly above the x-axis and partly below the x-axis. In that case, both the areas must be computed separately. The resultant area is then computed by adding up their moduli.

13 In case, the curve y = f(x) crosses the x-axis around n times when x varies from a to b, then the areas between y = f(x), the x-axis and the lines x = a and x = b is given by If the curve is symmetrical about the x-axis, or the y-axis, or both, then instead of computing the entire area, just the area of one of the symmetrical part can be found out calculate the area of one symmetrical part and multiply it by the number of symmetrical parts to get the whole area. Let f(x) be a continuous function in (a, b). Then the area bounded by the curve 1. y = f(x), the x-axis and the lines x = a and x = b is given by the formulae A = ab f(x) dx, provided f(x) > 0 (or f(x) < 0) x (a, b). 2. The area bounded by x = f(y), the y-axis and the lines y = c and y = d is given by A = cd f(y) dy, provided f(y) > 0 or f(y) < 0 y (c, d). 3. If we have two functions f(x) and g(x) such that f(x) < g(x) x [a, b], then the area bounded by the curves y = f(x), y = g(x) and the lines x = a, x = b (a < b) is given by A = ab [g(x) f(x)] dx.