5.7
Resolução de problemas: volume
O volume de um tanque de combustível montado na asa de uma aeronave a jato pode ser modelado pelo conceito de sólidos de revolução. Nesse caso, o tanq…
Um tanque de combustível montado na asa de uma aeronave a jato é formado girando uma região ao redor do eixo central. Essa região é formada girando uma função matemática ao redor do eixo x e se estende de zero a dois metros.
Para encontrar o volume do tanque, usa-se o método do disco, que envolve cortar o sólido em discos circulares infinitesimais e finos, perpendiculares ao eixo x.
Cada disco tem uma área igual a 𝜋 vezes o quadrado do valor da função. O volume total é encontrado integrando essas áreas ao longo do intervalo.
Após elevar a função ao quadrado, o integrando se simplifica para uma constante multiplicada pela potência dos segundos de x e a diferença entre dois e x.
Expandir e integrar essa expressão produz uma antiderivada envolvendo a terceira e a quarta potência de x.
Avaliando a integral definida de zero a dois e substituindo os limites, obtém-se uma expressão. Simplificando ainda mais, resulta em um volume de aproximadamente 1 metro cúbico, que é o volume total do tanque de combustível.
View the full transcript and gain access to JoVE Core videos
Q1: What is the disk method and how does it apply to finding volume?
The disk method divides a solid of revolution into infinitesimally thin circular disks perpendicular to the axis of rotation. Each disk's area equals π times the square of the function's radius value. By integrating these disk areas over the specified interval, you accumulate the total volume of the solid using calculation of volume of solids by integration.
Q2: How is a fuel tank modeled as a solid of revolution?
A fuel tank is modeled by rotating a two-dimensional region defined by a mathematical function about the x-axis. The region extends from zero to two meters along the axis, creating a three-dimensional symmetric shape. This approach allows engineers to calculate precise fuel capacity and weight distribution for aircraft performance.
Q3: What steps are involved in calculating the volume using integration?
First, square the function defining the tank's boundary to get the disk area formula. Then expand the resulting expression to identify terms with third and fourth powers of the variable. Finally, integrate these terms over the interval from zero to two meters and evaluate the definite integral by substituting the limits.
Q4: Why is the disk method appropriate for this fuel tank problem?
The disk method is appropriate because the boundary curve lies directly against the axis of rotation, creating circular cross-sections perpendicular to the x-axis. This geometry allows each disk's radius to equal the function value at that position, making the method straightforward and accurate for calculating the tank's volume.
Q5: What does the antiderivative represent in this volume calculation?
The antiderivative describes how volume accumulates along the axis of integration as you move from zero to two meters. It captures the cumulative effect of all infinitesimal disk areas stacked along the tank's length. Evaluating this antiderivative at the limits yields the total internal capacity of approximately one cubic meter.
Q6: How does the definite integral produce the final volume result?
The definite integral evaluates the antiderivative at the upper limit of two meters and subtracts its value at the lower limit of zero. This calculation yields a numerical result that simplifies to approximately one cubic meter, representing the total fuel tank capacity needed for aerospace engineering applications.
Q7: Why is precise volume calculation critical in aerospace engineering?
Accurate volume estimates determine fuel capacity, weight distribution, and overall aircraft performance. Engineers must know exact tank dimensions to optimize fuel storage, balance aircraft weight, and ensure safe flight operations. Mathematical modeling through integration provides the precision necessary for reliable aerospace design and safety.