Integrals

Integrals Exercise 1 : Proofs Prove $\int_{0}^{b} x^{3} dx = \frac{b^4}{4}$ using equal partitions and $\sum_{i=1}^{n} i^3$. Prove $\int_{0}^{b} x^{4} dx = \frac{b^5}{5}$ analogously. Show $\lim_{n \to \infty} \sum_{k=1}^{n} \frac{k^p}{n^{p+1}} = \frac{1}{p+1}$. Prove $\int_{0}^{b} x^p dx = \frac{b^{p+1}}{p+1}$. Exercise 2 For $0 < a < b$, find $\int_{a}^{b} x^p dx$ using partitions with fixed ratios $r = t_i/t_{i-1}$: Show $t_i = a \cdot c^{i/n}$ where $c = b/a$. For $f(x) = x^p$, derive: $$ U(f,P) = (b^{p+1} - a^{p+1}) \frac{c^{p/n}}{1 + c^{1/n} + \cdots + c^{p/n}} $$ and find $L(f,P)$. Conclude $\int_{a}^{b} x^p dx = \frac{b^{p+1} - a^{p+1}}{p+1}$. Exercise 3 Evaluate by symmetry: ...