let f:[a,b]Rbe a function.(a) Suppose that fis integrable ona,b. For ninN, define Pn={xj : Prove thatlimnR(f,Pn)=limnb-anj=1nf(xj)=abf(x)dx(b) Use part (a)to establish the integration formulaeabxdx=12(b2-a2) and abx2dx=13(b3-a3)(c) Part (a) above is useful for computing integrals provided that itis known in advance that fis integrable. Show that the existence alone oflimnR(f,Pn)on some interval is not sufficient for ftobe integrable on that interval.Let a let f:[a,b]->R be a function.(a) Suppose that f is integrable on a,b. For ninN, define P_(n)={(x_(j)):} : Prove that\lim_(n->\infty )R(f,P_(n))=\lim_(n->\infty )(b-a)/(n)\sum_(j=1)^n f(x_(j))=\int_a^b f(x)dx(b) Use part (a) to establish the integration formulae\int_a^b xdx=(1)/(2)(b^(2)-a^(2)) and \int_a^b x^(2)dx=(1)/(3)(b^(3)-a^(3))(c) Part (a) above is useful for computing integrals provided that it is known in advance that f is integrable. Show that the existence alone of \lim_(n->\infty )R(f,P_(n)) on some interval is not sufficient for f to be integrable on that interval.