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

Question

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

UpStudy AI · Accepted Answer

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:
  $$\frac{28}{3}$$
El valor de la integral $$ \int_{-2}^{2} f(x) \, dx $$ es $$ \frac{28}{3} $$ o $$ 9\frac{1}{3} $$.

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