Question

Double integrals are part of the fundamental concepts of Differential and Integral Calculus when we are interested in working with spatial notions of volumes or even surface areas. Based on this concept, judge the following information: I. To calculate a double integral in a rectangular region, we proceed with the use of dodecahedrons to approximate the volume of a surface. II. The gradient vector is used to calculate iterated integrals. III. The surface volume is approximated by the Riemann sum limit for functions of two variables. What is stated in:

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Answer to a math question Double integrals are part of the fundamental concepts of Differential and Integral Calculus when we are interested in working with spatial notions of volumes or even surface areas. Based on this concept, judge the following information: I. To calculate a double integral in a rectangular region, we proceed with the use of dodecahedrons to approximate the volume of a surface. II. The gradient vector is used to calculate iterated integrals. III. The surface volume is approximated by the Riemann sum limit for functions of two variables. What is stated in:

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As afirmações fornecidas referem-se a conceitos de cálculo multivariável, particularmente no que diz respeito a integrais duplas e suas aplicações. Vamos avaliar cada afirmação: I. Para calcular uma integral dupla em uma região retangular, não usamos dodecaedros. Integrais duplas são usadas para calcular volumes sob superfícies e são aproximadas pela soma dos volumes de prismas retangulares (ou às vezes cilindros em coordenadas polares) no processo limite, não de dodecaedros. Esta afirmação está incorreta. II. O vetor gradiente é um vetor de derivadas parciais que aponta na direção da maior taxa de aumento de uma função. Não é usado diretamente para calcular integrais iteradas. Integrais iteradas são geralmente calculadas usando antiderivadas em relação a uma variável de cada vez. Esta afirmação está incorreta. III. O volume da superfície, mais precisamente referido como o volume sob uma superfície, é de fato aproximado pelo limite da soma de Riemann para funções de duas variáveis. No contexto das integrais duplas, à medida que o número de subdivisões se aproxima do infinito, a soma de Riemann se aproxima do volume exato sob a superfície de uma determinada região. Esta afirmação está correta. Com base nas informações fornecidas, apenas a Afirmação III está correta. As afirmações I e II estão incorretas.

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