206 PRACTICAL MATHEMATICS 



(26) The curve y = a + be* passes through the points (0, 26-62), 

 (1, 35-70), and (2, 49-81). Find the values of a, b, and c. What 

 is the value of the area under the curve between x = and x = 1 ? 



(27) The curve y = x 2 1 is cut by the line y =x + 5. Find 

 the co-ordinates of the points of intersection and the area between 

 the line and the curve. 



(28) Find the area enclosed by the axes of reference and the 

 curve y = 3x 2 8x + 4. Find also the area enclosed by the 

 curve and the axis ot x. 



(29) Find the area enclosed by the two curves y 2 = 4# and 

 x* = 4,y. 



(30) Find the first two points at which the curve y = e? sin x 

 crosses the axis of x, and then find the area bounded by the curve 

 and the axis of x between these points. 



(31) Find the area of the loop of the curve y 2 = x 2 (x + l). 



(32) Find the area enclosed by the axes of reference and the 

 curve x = y 2 9y + 18. Find also the area enclosed by the 

 curve and the axis of y. 



(33) Find the area of the loop of the curve y 2 = x 2 (x + 4). 



(34) Find the area enclosed by the curve y = Wx 2 41# + 21 

 and the axes of reference. Find also the area enclosed by the 

 curve and the axis of x. 



(35) The curve y = 10 Vx rotates about the axis of x, generating 

 a surface of revolution. Find the volume between the sections 

 at x = 1 and x = 9. (B. of E., 1908.) 



(33) The curve y = 1 + Q-2x 2 rotates round the axis of x, gener- 

 ating a surface of revolution. What is the volume between the 

 sections at x = and x = 10 (B. of E., 1912.) 



(37) If the same part of the curve in Question 36 rotates about 

 the axis of y, what is the volume of the surface of revolution 

 generated ? 



(38) The curve y = ax n passes through the points (2, 7'46), 

 (4, 22'72). Find a and n. Let A be the area bounded by the curve, 

 the axis of x and the ordinates at x = 1 and x = 3, and let B be 

 the area bounded by the curve, the axis of y and the abscissae cor- 

 responding to the ordinates at x = 1 and x = 3. Find the volumes of 

 the surfaces of revolution generated by the area A rotating about 

 the axes of x and y respectively, and by the area B rotating 

 about the axes of x and y respectively. 



(39) The curve y = a+ bx 15 passes through the points (1, 1-82) 

 and (4, 5-32). Find a and b. Let the curve rotate about the 

 axis of x describing a surface of revolution. Find the volume 

 between the sections at x = 1 and x = 4. 



(40) The curve y = cu?* passes through the points (1, 3-5) and 

 (10, 12-6). Find a and b. The curve rotates about the axis of x, 

 describing a surface of revolution. Find the volume between the 

 sections at x = 1 and x = 10. (B. of E., 1913.) 



