Ejercicio 1
Utilice la suma de Riemann con n=6, para aproximar el área bajo la curva y=4-2x en el intervalo [0,3]. Utilice el extremo derecho.
$$A \thickapprox \sum_{i=1}^{n} f(c_{i}) \Delta{x} $$
$$\Delta{x} = \frac{b - a}{n} = \frac{3-0}{n} = \frac{3}{6} = \frac{1}{2}$$
$$c_{i} = a + i\Delta{x} = 0 + i\frac{1}{2} = \frac{1}{2} i = \frac{i}{2}$$
$$A \thickapprox \sum_{i=1}^{6} f(c_{i}) \Delta{x} $$
$$A \thickapprox \Delta{x} [f(c_{1}) + f(c_{2}) + f(c_{3}) + f(c_{4}) + f(c_{5}) + f(c_{6})]$$
$$ \thickapprox \frac{1}{2}[f(\frac{1}{2}) + f(\frac{2}{2}) + f(\frac{3}{2}) + f(\frac{4}{2}) + f(\frac{5}{2}) + f(\frac{6}{2})] $$
$$ \thickapprox \frac{1}{2}[f(\frac{1}{2}) + f(1) + f(\frac{3}{2}) + f(2) + f(\frac{5}{2}) + f(3)] $$
$$ \thickapprox \frac{1}{2} [(4-2(\frac{1}{2})) + (4 - 2(1)) + (4 - 2(\frac{3}{2})) + (4 - 2(\frac{5}{2})) + (4 - 2(3))] $$
$$ \thickapprox \frac{1}{2}[3 + 2 + 1 - 1 -2] $$
$$ \thickapprox \frac{1}{2}[3] = \color{green} \frac{3}{2} = 1.5 $$
Ejercicio 2
Utilice sumas de Riemann con n = 6 para aproximar el área bajo la curva y = 4-x² en el intervalo [0,3]. Utilice el extremo izquierdo.
$$A \thickapprox \sum_{i=1}^{n} f(x_{i} - 1) \Delta_{x}$$
$$x_{i} - 1 = a + (i-1) \Delta_{x} $$
$$\Delta_{x} = \frac{b-a}{n} = \frac{3-0}{6} = \frac{3}{6} = \frac{1}{2} $$
$$c_{i} = a + (i-1)\Delta_{x} = 0 + (i-1)\frac{1}{2} = \frac{1}{2}i - \frac{1}{2} $$
$$A \thickapprox \sum_{i=1}^{6} f(c_{i}) \Delta_{x} $$
$$A \thickapprox \Delta_{x}[f(c_{1}) + f(c_{2}) + f(c_{3})+ f(c_{4}) + f(c_{5}) + f(c_{6})] $$
$$A \thickapprox \frac{1}{2}[f(0) + f(\frac{1}{2}) + f(1) + f(\frac{3}{2}) + f(2) + f(\frac{5}{2})] $$
$$ Reemplazando f(x) = 4-x^{2} $$
$$ \frac{1}{2}[(4) + (\frac{15}{4}) + (3) + (\frac{7}{4}) + (0) + (\frac{-9}{4})] $$
$$ \frac{1}{2}[7 + \frac{13}{4}] \thickapprox[\frac{41}{4}] = \color{green} \frac{41}{8} = 5.13 $$
Ejercicio 3
Utilice sumas de Riemann con n = 8 para aproximar el área bajo la curva y = cos(x) en el intervalo [-π, π], nota: usando el extremo derecho
$$A \thickapprox \sum_{i=1}^{n} f(c_{i}) \Delta_{x} $$
$$A \thickapprox \frac{b-a}{n} = \frac{\pi - (-\pi)}{8} = \frac{2\pi}{8} = \frac{\pi}{4} $$
$$c_{i} = a + i\Delta_{x} $$
$$c_{i} = -\pi + \frac{\pi}{4}i $$
$$A \thickapprox \sum_{i=1}^{8} f(c_{i})\Delta_{x} $$
$$A \thickapprox \Delta_{x}[f(c_{1}) + f(c_{2}) + f(c_{3}) + f(c_{4}) + f(c_{5}) + f(c_{6}) + f(c_{7})+ f(c_{8})] $$
$$A \thickapprox \frac{\pi}{4}[f(\frac{-3\pi}{4}) + f(\frac{-\pi}{2}) + f(\frac{-\pi}{4}) + f(0) + f(\frac{\pi}{4}) + f(\frac{\pi}{2}) + f(\frac{3\pi}{4}) + f(\pi)] $$
$$A \thickapprox \frac{\pi}{4}[-0.707 + 0 + 0.707 + 1 + 0.707 + 0 - 0.707 - 1] $$
$$A \thickapprox \frac{\pi}{4}[0] $$
$$\color{green} A = 0 $$
Ejercicio 4
Utilice sumas de Riemann con n = 8, para aproximar el área bajo la curva y = 1/x en el intervalo [1,5]. Solución gráfica
Extremo derecho
$$A \thickapprox \sum_{i=1}^{n} f(c_{i})\Delta_{x} $$
$$\Delta_{x} \frac{b-a}{n} = \frac{5-1}{8} = \frac{4}{8} = \frac{1}{2} $$
$$c_{i} = a + i\Delta_{x} = 1 + i\frac{1}{2} = 1 + \frac{i}{2} $$
$$A \thickapprox \sum_{i=1}^{8} f(c_{i})\Delta_{x} $$
$$A \thickapprox \Delta_{x}[f(c_{1}) + f(c_{2}) + f(c_{3}) + f(c_{4}) + f(c_{5}) + f(c_{6}) + f(c_{7}) + f(c_{8})] $$
$$A \thickapprox \frac{1}{2}[f(\frac{3}{2}) + f(2) + f(\frac{5}{2}) + f(3) + f(\frac{7}{2}) + f(4) + f(\frac{9}{2}) + f(5)] $$
$$A \thickapprox \frac{1}{2}[\frac{2}{3} + \frac{1}{2} + \frac{2}{5} + \frac{1}{3} + \frac{1}{3} + \frac{2}{7} + \frac{1}{4} + \frac{2}{9} + \frac{1}{5}] $$
$$A \thickapprox \frac{1}{2}[1 + \frac{3}{5} + \frac{2}{7} + \frac{3}{4} + \frac{2}{9}] $$
$$A \thickapprox \frac{1}{2}[2.8579] $$
$$\color{green} A \thickapprox 1.42895 \thickapprox 1.43 $$
Extremo izquierdo