AREAS 197 



Example 2. Find the first two points at which the curve 

 y - t~* sin x crosses the axis of x, and then find the area bounded 

 by the curve and the axis of x between these points. 



Now sin x = when x has the values 0, TT, 27C, etc., and there- 

 fore y = when x has these values. 



The first two points at which the curve crosses the axis of x 

 occur when x = and when x TT. 



!> 



Then the area = I y dx 

 o 



e~* sin x dx. 







To integrate e~* sin x we must integrate by parts. 



I 



sin x dx = uv \v du 



where u = sin x, du= cos x dx 

 and dv = e~* dx, v = e~* 



Then 



I -* sin x dx = e~* sin x + \e~* cos a; dx . . . (1) 



Also I e~* cos a? da? = uv \vdu 



where w = cos x, du = sin x dx 

 and du = e~* dx, v = e"* 



Then I e"* cos x dx = e~* cos x I e~* sin x dx . . . (2) 

 Solving (1) and (2) for U~" sin a; cte 

 I e~* sin a? da; = <?"* sin a; e~~* cos u 



sn a; 



f 1 



and I e~* sin a; da; = z(e~ x sin a? + e~* cos a;} 



If "I* 



Then area = - e~* sin x + e~ x cos x 



T{*~* COS 7C 1} 



2 



0-5216 



