Riemann Sums and Definite Integrals — Basic Exercises

1Problem 1

For an arbitrary partition $0=x_0<\cdots <x_n=1$0=x_0<\cdots<x_n=1 and sample points ${\xi }_i\in [x_{i-1},x_i]$\xi_i\in[x_{i-1},x_i], prove that the Riemann sum converges to $1/2$1/2, and hence evaluate $${\int }_0^{1}xdx$$\int_0^1x\,dx.

1.1Sample Solution

Let $m_i=(x_{i-1}+x_i)/2$m_i=(x_{i-1}+x_i)/2 and $\mathrm{\Delta }{x}_i=x_i-x_{i-1}$\Delta x_i=x_i-x_{i-1}. Then

Because $|{\xi }_i-m_i|\text{[PARSE ERROR: Undefined("Command("le")")]}\mathrm{\Delta }{x}_i/2$|\xi_i-m_i|\le\Delta x_i/2,

Thus the limit is independent of the partition and sample points, and $${\int }_0^{1}xdx=1/2$$\int_0^1x\,dx=1/2.

1.2Explanation

The estimate $\left|P\right|/2\to 0$|P|/2\to0 proves convergence for arbitrary partitions and sample points, not merely for one particular sequence of partitions.

2Problem 2

Evaluate $${\int }_{-2}^{2}xdx$$\int_{-2}^{2}x\,dx and the corresponding geometric area.

2.1Sample Solution

The definite integral is $0$0. The geometric area is $${\int }_{-2}^2\left|x\right|dx=4$$\int_{-2}^{2}|x|\,dx=4.

2.2Explanation

A definite integral represents signed area.

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