\subsection*Problem 4 Evaluate ( \int_0^1 x e^x^2,dx ) using substitution.

# Riemann Integral: Problems and Solutions Problem 1 Compute the Riemann sum for f(x) = x² on [0,2] using 4 subintervals and right endpoints.

\subsection*Solution 3 No. For any partition, upper sum (U(P,f)=1) (since every interval contains rationals), lower sum (L(P,f)=0) (since every interval contains irrationals). Thus (\inf U \neq \sup L), so (f) is not Riemann integrable.

\subsection*Solution 9 Since (f \ge 0), any lower sum (L(P,f) \ge 0). The integral is the supremum of lower sums, hence (\int_a^b f = \sup L(P,f) \ge 0).