44. \( \int_{-2}^{2} f(x) d x \quad \) donde \( f(x)=\left\{\begin{array}{ll}2 & \text { si }-2 \leqslant x \leqslant 0 \\ 4-x^{2} & \text { si } 0

Ask by Norris Todd. in Colombia

Mar 31,2025

La función \( f(x) \) está definida en dos intervalos. Por lo tanto, podemos dividir la integral en dos partes: \[ \int_{-2}^{2} f(x) \, dx = \int_{-2}^{0} 2 \, dx + \int_{0}^{2} (4 - x^{2}) \, dx \] Calculemos la primera integral: \[ \int_{-2}^{0} 2 \, dx \] Evaluate the integral by following steps: - step0: Solution: \(\int_{-2}^{0} 2 dx\) - step1: Evaluate the integral: \(\int 2 dx\) - step2: Evaluate the integral: \(2x\) - step3: Return the limits: \(\left(2x\right)\bigg |_{-2}^{0}\) - step4: Calculate the value: \(4\) Ahora calculemos la segunda integral: \[ \int_{0}^{2} (4 - x^{2}) \, dx \] Evaluate the integral by following steps: - step0: Evaluate using formulas and rules: \(\int_{0}^{2} \left(4-x^{2}\right) dx\) - step1: Evaluate the integral: \(\int \left(4-x^{2}\right) dx\) - step2: Use properties of integrals: \(\int 4 dx+\int -x^{2} dx\) - step3: Evaluate the integral: \(4x+\int -x^{2} dx\) - step4: Evaluate the integral: \(4x-\frac{x^{3}}{3}\) - step5: Return the limits: \(\left(4x-\frac{x^{3}}{3}\right)\bigg |_{0}^{2}\) - step6: Calculate the value: \(\frac{16}{3}\) Los resultados de las integrales son: 1. \(\int_{-2}^{0} 2 \, dx = 4\) 2. \(\int_{0}^{2} (4 - x^{2}) \, dx = \frac{16}{3}\) Ahora sumamos ambos resultados: \[ 4 + \frac{16}{3} \] Calculate the value by following steps: - step0: Calculate: \(4+\frac{16}{3}\) - step1: Reduce fractions to a common denominator: \(\frac{4\times 3}{3}+\frac{16}{3}\) - step2: Transform the expression: \(\frac{4\times 3+16}{3}\) - step3: Multiply the numbers: \(\frac{12+16}{3}\) - step4: Add the numbers: \(\frac{28}{3}\)