b 2. Let f(x) = cc R for all x ( [a, b]. Show that f is integrable and find / f(x)dx using the definition of the Riemann integral.Given that : f ( x ) = c ERR + x E[a, b]. We have to show that of is integrable and we have to find ( free ) do using the definition a of the Riemann integral. Proof : Since f ( x ) = c, + xe[a,b] where c ER is a real number. Here of is a constant function. Since, f is a constant function, so by definition it is bounded on [ a , b ] . Now, of is a constant function and bounded so its supremum and infimum exists, and both are equal to c. ise. Sup f = inff = c. Let P = ( a = 20, 21 , *2 , ...., kn = by be a partition on [a,b]. Let my and My be the infimum and supsumum of f in with subinterval denoted by In = [ xx, , 207 Since f(x ) = c , + xE [aib] and of is a constant function. => my = c and My = C.Now, Lower Riemann Sum is given by : n L ( P, f ) = Z My fx = CS dy n 9 =1 = c (b-a ) : Length of subinterval = b-al Similarly , Upper Riemann sum is given by : n U ( P,f ) = 5 Mady = cs d M = 1 = c (b - a). Now, a f ( x ) dx = sup & L ( P , A ) : Ped Caibly = c (b-a) b and f (x ) do = influ ( P . f ) : P E $ [a , bly a = c ( b - a ) Here , we found that: b b ( f (x ) dx = / f(x ) dx = c ( b-a) . a a Hence , we can say that : 6 f is integrable over [a, b ) and ( f( x ) dx = c ( b- a ) a Proved