Question

solid obtained by rotation around the axis x = -1, the region delimited by x^2 - x + y = 0 and the abscissa axis

259

likes
1294 views

Answer to a math question solid obtained by rotation around the axis x = -1, the region delimited by x^2 - x + y = 0 and the abscissa axis

Expert avatar
Eliseo
4.6
63 Answers
Para resolver esse problema, podemos primeiro determinar os limites de integração para a rotação em torno do eixo x = -1.

A equação x^2 - x + y = 0 pode ser reescrita como y = x - x^2.

A região delimitada por essa equação e o eixo das abscissas é representada pelo gráfico abaixo:

\begin{tikzpicture}\begin{axis}[ axis lines = left, xlabel = $x$, ylabel = $y$, xmin=-1, xmax=1, ymin=0, ymax=1,]\addplot [ domain=-1:1, samples=100, color=blue,]{x-x^2};\end{axis}\end{tikzpicture}

Agora, para obter o sólido obtido pela rotação em torno do eixo x = -1, podemos usar o método do disco ou do anel.

Vamos usar o método do anel.

O raio do anel é dado pela distância entre o ponto (x, x-x^2) no gráfico e o eixo de rotação x = -1, que é r(x) = x - (-1) = x + 1.

A área do anel é dada por A(x) = π[r(x)^2 - r(x - \Delta x)^2], onde \Delta x é uma pequena variação no valor de x.

Agora, vamos calcular o volume do sólido integrando as áreas dos anéis ao longo do intervalo de x.

[passar o texto]

Portanto, o volume do sólido é dado por:

V = \int_{a}^{b} A(x) dx

V = \int_{-1}^{1} \pi[x + 1)^2 - (x + \Delta x + 1)^2] dx

V = \int_{-1}^{1} \pi[(2x + 1) \Delta x - \Delta x^2] dx

V = \pi \int_{-1}^{1} (2x + 1) \Delta x - \pi \int_{-1}^{1} \Delta x^2 dx

Agora, vamos resolver cada umas das integrais.

\pi \int_{-1}^{1} (2x + 1) \Delta x = \pi [\Delta x^2 + x^2 + x] \bigg|_{-1}^{1}

\pi [\Delta x^2 + 1 + 1] - \pi [\Delta x^2 + 1 - 1]

\pi [\Delta x^2 + 2] - \pi [\Delta x^2]

2 \pi

A segunda integral é:

\pi \int_{-1}^{1} \Delta x^2 dx = \pi \Delta x^3/3 \bigg|_{-1}^{1}

\pi [(1/3) - (-1/3)]

\pi (2/3)

Agora, vamos substituir esses valores na fórmula do volume:

V = 2 \pi - \pi (2/3)

V = 2 \pi - (2/3) \pi

V = (2 - 2/3) \pi

V = (4/3) \pi

Portanto, a resposta é:

\text{Answer: } V = (4/3) \pi

Frequently asked questions (FAQs)
What is the value of sin(45°) + cos(60°) - tan(30°) multiplied by cot(45°)?
+
Math Question: Find the absolute extrema of the function f(x) = x^2 - 6x + 9 in the interval [0, 10].
+
What value of 'c' in the function f(x) = c will make the graph of the function a horizontal line?
+
New questions in Mathematics
To calculate the probability that a player will receive the special card at least 2 times in 8 games, you can use the binomial distribution. The probability of receiving the special card in a single game is 1/4 (or 25%), and the probability of not receiving it is 3/4 (or 75%).
Revenue Maximization: A company sells products at a price of $50 per unit. The demand function is p = 100 - q, where p is the price and q is the quantity sold. How many units should they sell to maximize revenue?
what is 456456446+24566457
4x567
Suppose 56% of politicians are lawyers if a random sample of size 564 is selected, what is the probability that the proportion of politicians who are lawyers will differ from the total politicians proportions buy more than 4% round your answer to four decimal places
How many anagrams of the word STROMEC there that do not contain STROM, MOST, MOC or CEST as a subword? By subword is meant anything that is created by omitting some letters - for example, the word EMROSCT contains both MOC and MOST as subwords.
Calculate the value of a so that the vectors (2,2,−1),(3,4,2) and(a,2,3) are coplanar.
The market for economics textbooks is represented by the following supply and demand equations: P = 5 + 2Qs P = 20 - Qd Where P is the price in £s and Qs and Qd are the quantities supplied and demanded in thousands. What is the equilibrium price?
How much does the average college student spend on food per month? A random sample of 50 college students showed a sample mean $670 with a standard deviation $80. Obtain the 95% confidence interval for the amount college students spend on food per month.
Professor Vélez has withdrawn 40 monthly payments of $3,275 from her investment account. If the investment account yields 4% convertible monthly, how much did you have in your investment account one month before making the first withdrawal? (Since you started making withdrawals you have not made any deposits.)
Next%C3%B3n%2C+we+are+given+a+series+of+Tri%C3%A1angles+Right%C3%A1angles+%3Cbr%2F%3Ey+in+each+one+of+them+ are+known+2%28two%29+measurements+of+sides.+%3Cbr%2F%3Elet's+determine+all+trigonom%C3%A9tric+ratios.
cube root of 56
Determine the Linear function whose graph passes through the points (6, -2) and has slope 3.
The blood types of individuals in society are as follows: A: 30%, B: 25%, AB: 20%, 0: 25%. It is known that the rates of contracting a certain disease according to blood groups are as follows: A: 7%, B: 6%, AB: 7%, 0: 4%. Accordingly, if a person selected by chance is known to have this disease, what is the probability of having blood group O?
If sin A=0.3 and cos A=0.6, determine the value of tan A.
A post office has three categories of letters: 60% are from businesses, 30% are individual mail, and the remaining 10% are government mail. 5% of the letters from businesses have address errors, 10% of the individual mail has address errors, while 1% of the government mail has address errors. If we receive a letter with an address error, what is the probability that it is individual mail?"
Write the inequality in the form of a<x<b. |x| < c^2
a coffee shop has 9 types of creamer and 11 types of sweetener. In how any ways can a person make their coffee?
3(x-4)=156
Sarah is lining a square tray with 1 inch square tiles. the side length of the tray is 9 inches. How many tiles does Sarah need?