Areas of Bounded Regions Notes

AREAS OF BOUNDED REGIONS APPLICATIONS OF INTEGRALS AREAS OF BOUNDED REGIONS AREA UNDER SIMPLE CURVE 1. The area of the region bounded by a curve y = f(x), x =a, x = b, and the x - axis. Case – I When the curve y = f(x) lies above the x-axis. Y y = f(x) B The area bounded by the curve y = f(x), the x-axis A b  and the ordinates x = a and x = b is given by y dx a C X’ O D b X b B X a Y’ Case – II When the curve y = f(x) lies below the x-axis. Y O X’ C a A The area bounded by the curve y = f(x), the x-axis, b and the ordinaes x = a and x = b is given by  ( y) dx a 2. y=d B Case – I When the curve x = f(y) lies to the right of the y -axis. The area bounded by the curve x = f(y), the y-axis d  C y=c X’ x dy X Y’ Case – II When the curve x = f(y) lies to the left of the y-axis. The area bounded by the curve x = f(y), the y-axis y=d x=f(y) y=c d  D x=f(y) A c and the absissa y = c and y = d is given by D Y The Area of the Region Bounded by a Curve x = f(y), The Abscissae y = c, y = d and the y -axis. and the abscissae y = c and y = d is given by C Y’ y=f(x) Y (  x ) dy X’ O c Y’ Y The Area Bounded by Curve y = f(x), Ordinates x = a and x = b and x -axis is given by A=  a b ydx +  (-y)dx , where a < c < b. X c y = f(x) O 3. X c Y’ x=a x=c x=b Y The Area Between Two Curves Case – I The area bounded by two curves y = f(x) and y = g(x), which are intersected by the ordinates x = a and y=f(x) y=g(x) b  [f ( x)  g( x)]dx X x = b is given by O 4. a Y’ a b X 6.1 X AREAS OF BOUNDED REGIONS Case – II The area bounded by two curves y = f (x) Y y=f(x) b and y = g(x) is given by  {f ( x )  g( x )}dx , a y=g(x) X’ O of the two curves. x=b x=a X where a and b are the abscissae of the points of intersection Y’ Case – III The area bounded by two curves x = f(y) Y d  {f (y )  g( y )}dx , y=d O B(0,a) a a X’ Y’ Y x=a X’ Latus rectum F X O F X’ Latus rectum x=–a L’ L’ Y’ y2 = 4ax, a > 0 Y y Latus rectum It is symmetric about y-axis, where LL is the latus rectum. y=b F L’ O X’ y=–b O X 3. Ellipse = 4by, b > 0 x2 y2   1 represents an ellipse with centre at (0, 0). a2 b2 It is symmetric about both the axes, meeting x-axis at (±a, 0) and y-axis at (0, ±b) and a > b. Here, AA = 2a = length of major axis. BB = 2b = length of minor axis. F L Latus rectum Y’ x2 = 4by, b < 0 Y’ x2 X L L’ X’ X O Y’ x2 = 4by, where b > 0 or x2 = 4by, where b < 0 O is the vertex, F is the focus, LL is the latus rectum, LL'  YY' . X A(a,0) O (0,0) L LL'  X' X; X' X being the axis of the parabola. (ii) Y Y g2  f 2  c Parabola y2 = 4ax, where a > 0 or y2 = 4ax, whre a < 0 (standard equation) It is symmetric about x -axis, where O is the vertex, F is the focus, LL is the latus rectum : X Y’ SOME STANDARD CURVES 1. Circle (i) x2 + y2 = a2 represents a circle with centre at (0,0) and radius = a. It is symmetric about both the axes. (ii) (x–)2 + (y – )2 = a2 represents a circle with centre at (, ) and radius = a. (iii) x2 + y2 + 2gx + 2fy + c = 0 represents the general equation of a circle with centre at (–g, – f ) and radius 2. (i) x X’ of the two curves. = = y=c where c and d are ordinates of the points of intersection 6.2 (h)g= x c )f (h and x = g(y) is given by Y B(0,b) A’(–a,0) X’ A(a,0) O (0,0) B’(0,–b) Y’ X