Application of Integrals - Mathematics, Study notes of Mathematics

Download Application of Integrals - Mathematics and more Study notes Mathematics in PDF only on Docsity! 8.1 Overview This chapter deals with a specific application of integrals to find the area under simple curves, area between lines and arcs of circles, parabolas and ellipses, and finding the area bounded by the above said curves. 8.1.1 The area of the region bounded by the curve y = f (x), x-axis and the lines x = a and x = b (b > a) is given by the formula: Area = ∫ b a ydx = ( )∫ b a f x dx 8.1.2 The area of the region bounded by the curve x = (y), y-axis and the lines y = c, y = d is given by the formula: Area = ( ) d d c c xdy y dy=∫ ∫ 8.1.3 The area of the region enclosed between two curves y = f (x), y = g (x) and the lines x = a, x = b is given by the formula. Area = [ ]( ) – ( ) b a f x g x dx∫ , where f (x) ≥ g (x) in [a, b] 8.1.4 If f (x) ≥ g (x) in [a, c] and f (x) ≤ g (x) in [c, b], a < c < b, then Area = [ ] ( )( ) – ( ) ( ) – ( ) c b a c f x g x dx g x f x dx+∫ ∫ 8.2 Solved Examples Short Answer (S.A.) Example 1 Find the area of the curve y = sin x between 0 and π. Solution We have Area OAB = = 0 – cos x  = cos0 – cosπ = 2 sq units. Chapter 8 APPLICATION OF INTEGRALS APPLICATION OF INTEGRALS 171 Example 2 Find the area of the region bounded by the curve ay2 = x3, the y-axis and the lines y = a and y = 2a. Solution We have Area BMNC = 2 2 1 2 3 3 a a a a xdy a y dy=∫ ∫ = 1 253 3 3 5 a a a y = ( ) 1 553 33 3 2 – 5 a a a = ( ) 1 5 5 3 3 3 3 2 – 1 5 a a = 2 2 33 2.2 – 1 5 a sq units. Example 3 Find the area of the region bounded by the parabola y2 = 2x and the straight line x – y = 4. Solution The intersecting points of the given curves are obtained by solving the equations x – y = 4 and y2 = 2x for x and y. We have y2 = 8 + 2y i.e., (y – 4) (y + 2) = 0 which gives y = 4, –2 and x = 8, 2. Thus, the points of intersection are (8, 4), (2, –2). Hence Area = 4 2 –2 14 – 2  +  ∫ y y dy = 42 3 –2 14 – 2 6 y y y+ = 18 sq units. Example 4 Find the area of the region bounded by the parabolas y2 = 6x and x2 = 6y. 174 MATHEMATICS From Fig. 8.8 area ODAB = ( )2 0 2 – –∫ a ax x ax dx Let x = 2a sin2θ. Then dx = 4a sinθ cosθ dθ and x = 0, ⇒ θ = 0, x = a ⇒ θ = 4  . Again, 2 0 2 – a ax x dx∫ = ( ) ( ) 4 0 2 sinθcos sin cosa a dθ 4 θ θ θ∫  = a2 ( ) 4 4 00 sin 41– cos4 – 4 d a2 θ θ θ = θ  ∫   = 4  a2 . Further more, 0 a ax dx∫ = 3 2 0 2 3 a a x       = 22 3 a Thus the required area = 2 2π 2– 4 3 a a = a2 2– 4 3      sq units. Example 9 Find the area of a minor segment of the circle x2 + y2 = a2 cut off by the line x = 2 a . Solution Solving the equation x2 + y2 = a2 and x = 2 a , we obtain their points of intersection which are , 3 2 2 a a     and 3, – 2 2 a a     . APPLICATION OF INTEGRALS 175 Hence, from Fig. 8.9, we get Required Area = 2 Area of OAB = 2 2 2 2 – a a a x dx∫ = 2 2 2 2 –1 2 – sin 2 2   +    a a x a x a x a = 2 2 2π 3 π. – . – . 2 2 4 2 2 6       a a a a `= ( ) 2 6π –3 3 – 2π 12 a = ( ) 2 4π –3 3 12 a sq units. Objective Type Questions Choose the correct answer from the given four options in each of the Examples 10 to 12. Example 10 The area enclosed by the circle x2 + y2 = 2 is equal to (A) 4π sq units (B) 2 2π sq units (C) 4π2 sq units (D) 2π sq units Solution Correct answer is (D); since Area = 2 2 0 4 2 – x∫ = 4 2 2 –1 0 2 – sin 2 2 x x x  +   = 2π sq. units. Example 11 The area enclosed by the ellipse 2 2 2 2 x y a b + = 1 is equal to (A) π2ab (B) πab (C) πa2b (D) πab2 Solution Correct answer is (B); since Area = 4 2 2 0 – a b a x dx a∫ 176 MATHEMATICS = 2 2 2 –1 0 4 – sin 2 2 a b x a x a x a a   +    = πab. Example 12 The area of the region bounded by the curve y = x2 and the line y = 16 (A) 32 3 ` (B) 256 3 (C) 64 3 (D) 128 3 Solution Correct answer is (B); since Area = 2 16 0 ydy∫ Fill in the blanks in each of the Examples 13 and 14. Example 13 The area of the region bounded by the curve x = y2, y-axis and the line y = 3 and y = 4 is _______. Solution 37 3 sq. units Example 14 The area of the region bounded by the curve y = x2 + x, x-axis and the line x = 2 and x = 5 is equal to ________. Solution 297 6 sq. units 8.3 EXERCISES Short Answer (S.A.) 1. Find the area of the region bounded by the curves y2 = 9x, y = 3x. 2. Find the area of the region bounded by the parabola y2 = 2px, x2 = 2py. 3. Find the area of the region bounded by the curve y = x3 and y = x + 6 and x = 0. 4. Find the area of the region bounded by the curve y2 = 4x, x2 = 4y. 5. Find the area of the region included between y2 = 9x and y = x 6. Find the area of the region enclosed by the parabola x2 = y and the line y = x + 2 7. Find the area of region bounded by the line x = 2 and the parabola y2 = 8x 8. Sketch the region {(x, 0) : y = 24 – x } and x-axis. Find the area of the region using integration. 9. Calcualte the area under the curve y = 2 x included between the lines x = 0 and x = 1. 10. Using integration, find the area of the region bounded by the line 2y = 5x + 7, x- axis and the lines x = 2 and x = 8.