Sample problems on Area under a curve - Application of Integrals

Area under the curve- Sample Problems Find the area of shaded region shown below. The equation of curve is Given Area below the curve To find the area of the shaded region, we need to find the Area below the line (lined joined by point (1, 1) and (3,) ) and subtract the area under the curve Area under the line = = > So Area of shaded region = This is similar to last problem... First, find the area under the line AP and then subtract the area under the curve to find the area of shaded region. Area under the line AP = Area of triangle APB = Area under the curve A= 192 Area of shaded region = 288 – 192 = 96 Ans: 96 forward.. = Area from Area from = = 1 -2 = -1 Since area cannot be negative, it will be taken as 1 Area from Since area from Either Since the plot is in the positive x direction, Ans: a = 2 Area A = We need to find the coordinate Q. For this we need to find the equation of tangent at (3, 27) Slope of the Tangent line = = at (3, 27) = Coordinates of point Q can be found by putting y = 0 Area under the Tangent PQ and X axis = Area of the triangle PQ3 = For area of shaded region in the problem, subtract the area under tangent from the area under PO PS: The problem asks area under the tangent, so you need not do the last step. However, if you are supposed to find the area of shaded region, you need to do the last step... First you need to find the point P. Since this is the point of minima, coordinates of P would be (2, 1). Now we need to find equation of line passing through point P (2,1) and slope 2 (Equation of tangent line) Now we also need the coordinates of point Q. That can be found by finding the intersection point of tangent line and the curve. Intersection point can be found easily.. Area of curve 6 Area of curve 24 +10 14 So, area of shaded region= Ans: 4/3 Find the area of the region bounded by tangent to the curve Equation of Tangent passing through (1, 1) Y- 1 = n (x-1) This tangent cuts the x axis at (put y=0 in the equation of Tangent) 0-1 = n(x-1) => x = 1-1/n Area of the region bounded by the tangent and X=1 is given by = = (1 + n/ 2 – n) - [+ n = 1/2n 1/) (1, 1/) (1, 1)