RVHS H2 Math Applications of Integration Qns

River Valley High School, Mathematics Department 9758 H2 Mathematics 2026 Applications of Integration 1 Applications of Integration 1. CJC/2009/II/2 (a) Calculate the area of the region bounded by the curves ( ) 1e 4 −= −xy and ( ) 1ln −= xy . [3] (b) The region S is bounded by the curve 12 += xy and the lines 1=y and bx= , where 0b . xV is the volume of the solid of revolution formed when S is rotated completely about 1=y . yV is the volume of the solid of revolution when S is rotated completely about the y-axis. Find the value of b such that yx VV = . [6] 2. IJC/II/3 The diagram shows the region R bounded by the curve 7e e 3e x xxy −= + , the y-axis, and the line OA, where A is the point on the curve when ln 2x= . A O R y x 12+= xyb 0 1=y S

River Valley High School, Mathematics Department 9758 H2 Mathematics 2026 Applications of Integration 2 (i) By means of the substitution exu= , find the exact value of ln 2 0 e d e 3e x xx x−+ . [4] (ii) Calculate the exact area of the region R. [2] (iii) Find the numerical value of the volume of the solid formed when R is rotated through 4 right angles about the x-axis. [3] (iv) Find the equation of the curve obtained when the curve 7e e 3e x xxy −= + is translated 5 units in the negative direction of the y-axis, followed by a stretch parallel to the x-axis with a scale factor of 3. [2] 3. PJC/I/9 The diagram shows the parts of the graph of 2 1y x= + between 1x= and 2x= . Prove that the total area of the four rectangles, each of equal width, as shown in the diagram below, may be expressed as 4 1 2 8r r= + . [2] In general, when there are n rectangles, each of equal width, under the curve between 1x= and 2x= (instead of just 4), find an expression, in the form ( ) 1 f n r r =  , for their total area. Hence show that 1 23 2ln22 n r nr=  +  . Deduce that 11 2 3 2 2ln2 2 2 1 nn rr n r n r==  + + −  . 4. ACJC/2010/I/11 The region R in the first quadrant is bounded by the curve ( )222y a a x=− , where 0a , and the line joining ( )2 ,0a and ( )0,a . The region S, lying in the first quadrant, is bounded by the curve ( )222y a a x=− and the lines 2xa= and ya= . (i) Draw a sketch showing the regions R and S. [1] (ii) Find, in terms of a, the volume of the solid formed when S is rotated completely about the x- axis. [4] (iii) By using a suitable translation, find, in terms of a, the volume of the solid formed when R is rotated completely about the line 2xa= . [4] x O y 1 2

River Valley High School, Mathematics Department 9758 H2 Mathematics 2026 Applications of Integration 3 5. AJC/2011/I/4 (i) By using the substitution tantx= , show that ( ) 1 2 11 d tan 2 tan1 sin 2 x x cx −=++ . [3] (ii) Find the exact volume of revolution when the region bounded by the curve 2 12, 1 siny x=+ + the lines 4x = , y = 2 and the y-axis is rotated completely about the line y = 2. [3] 6. IJC/2011/I/11 (i) Verify, without the use of a calculator, that is a root of the equation . [1] (ii) By sketching the graphs of and , solve exactly the inequality for . [2] (iii) Hence evaluate exactly 2 3 22 0 tan 2cos d22 xx x     −       . [6] 7. PJC/2011/II/5 (i) Find 21d 4 4sind2 xxxx − −+   , giving your answer in its simplest form. [3] (ii) Hence, or otherwise, show that 2 2 1 0 1 4 d 4 sin22 k kx x a k − − = − +  where a is to be found in terms of k, 0 < k < 2. [2] (iii) Sketch the curve C with equation 2244yx+= and show by shading, a region whose area is given by the integral in (ii). [2] (iv) Hence, find the area of the region lying inside C and between the lines x = –1 and x = 1, giving your answer in exact form. [3] 8. CJC/2013/I/10 (a) Use the substitution 2secx = to find 22 1 d 4 x xx − . (b) The region bounded by the curve 2 1 12 y x = + , the x-axis, the lines 3 2x=− and 1 2 x=− is rotated completely about the x-axis to form a solid of revolution of volume V. Find the exact value of V, giving your answer in the form 2k . 4  = 22tan 2cos 0−= 2tan 2 xy =  22cos 2 xy =  22tan 2cos22 xx          x−  

River Valley High School, Mathematics Department 9758 H2 Mathematics 2026 Applications of Integration 4 9. MJC/2013/I/9 (a) Find the exact value of p such that 1 2201 2e1 d ln d1 p x x xpx =+ . [5] (b) The curve C has equation 21 xy x= + . (i) Sketch the curve C. [1] (ii) Use the substitution 2ux= to find ( ) 220 d 1 n x x x+ , for 0n . [3] (iii) Hence find the exact volume of revolution formed when the region between the curve and the positive x-axis is rotated completely about the x-axis. [2] 10. HCI/2014/I/11 A curve has equation given by ( ) 2 ln 1yx=− , where 0x . (i) Sketch the graph, indicating the exact coordinates of the x -intercepts and the turning point. [4] The region R is bounded by the curve and the -axis. (ii) Find the exact area of R . [4] (iii) Find the volume of the solid generated when R is rotated through 2 radians about the y- axis. [4] 11. TPJC/2014/II/2 A curve C has equation . (i) Using an algebraic method, find the range of values of y for which C does not exist. [3] (ii) Sketch C, stating the exact coordinates of any points of intersection with the axes, the exact coordinates of turning points and the equations of any asymptotes. [3] (iii) The region enclosed by C, the lines 𝑥 = −2 and 4y  is denoted by R. Find the numerical value of the volume of revolution when region R is rotated through 2 radians about the x- axis. [3] x 2 4 12 3 xxy x − + += +

River Valley High School, Mathematics Department 9758 H2 Mathematics 2026 Applications of Integration 5 12. JJC/2014/I/4 The diagram shows a region R in the first quadrant bounded by the curve C with equation 2 2 3 4 y x =+ − , the y-axis and the line y = 5. The line y = 5 and the curve intersect at the point ( )3,5 . (i) Calculate the area of region R. [2] (ii) Write down the equation of the curve obtained when C is translated by 5 units in the negative y-direction. [1] (iii) Hence show that the volume of the solid formed when R is rotated completely about the line 5y= is given by 3 2 2 0 124 1 d 4 4 xx x   −+ − − , and evaluate this integral exactly. [4] 13. MJC/2014/I/9 It is given that ( ) 1f2 1x x=− − . (i) On separate diagrams, sketch the graphs of ( )fyx= and ( )fyx= , giving the coordinates of any points where the graphs meet the x- and y- axes. You should label the graphs clearly. [4] (ii) Use a non-calculator method to find the area bounded by ( )fyx= , 5 4x= , 2x= and the x- axis. [4] (iii) The region R is bounded by ( )fyx= , 2y= and 3 2x= . Find the volume of revolution formed when R is rotated completely about the y-axis. [4] x y = 5 y O R C x = 2

River Valley High School, Mathematics Department 9758 H2 Mathematics 2026 Applications of Integration 6 14. RJC/2014/I/7 Curves 1C and 2C are given by the equations 22 1 22 2 : 25 : 100 C x y C ax by += += where a and b are positive real constants such that ab . (a) Find the condition(s) on the values of a and b such tha