Question

Calculate ∬_RdA where R is the region bounded in the first quadrant by y^2=x^3 and the line y=x.

153

likes
763 views

Answer to a math question Calculate ∬_RdA where R is the region bounded in the first quadrant by y^2=x^3 and the line y=x.

Expert avatar
Clarabelle
4.7
91 Answers
¡Absolutamente! Aquí se explica cómo calcular la integral doble y encontrar el área de la región. **1. Dibuja la región** Primero, siempre es una buena práctica visualizar la región de integración. * **y^2 = x^3:** Esta es una parábola lateral que se abre hacia la derecha. * **y = x:** Esta es una línea recta que pasa por el origen con una pendiente de 1. Se cruzan en el primer cuadrante formando la región R. **2. Determinar los límites de la integración** Dado que estamos tratando con una región algo inusual, es más fácil integrar primero con respecto a 'y' y luego a 'x' (dy dx). * **límites de y:** y va desde 0 hasta el punto donde se cruzan la línea y la curva. Para encontrar este punto, sustituye y=x en la ecuación y^2 = x^3. Esto nos da x^2 = x^3 => x = 1 (descartamos x = 0 ya que es el origen). Por tanto, 0 ≤ y ≤ 1. * **límites de x:** Para cada valor de y, x va desde la recta y=x hasta la curva y^2 = x^3. Resolviendo la ecuación de la curva para x, obtenemos x = y^(2/3). Entonces, y ≤ x ≤ y^(2/3). **3. Configurar la integral doble** La integral doble que representa el área es: ∬_RdA = ∫_(0)^(1) ∫_(y)^(y^(2/3)) dx dy **4. Evaluar la integral interna** ∫_(y)^(y^(2/3)) dx = [x]_(y)^(y^(2/3)) = y^(2/3) - y **5. Evaluar la integral exterior** ∫_(0)^(1) (y^(2/3) - y) dy = [3/5 * y^(5/3) - 1/2 * y^2 ]_(0)^(1 ) = 3/5 - 1/2 = 1/10 **6. El resultado** El valor de la integral doble es 1/10 unidades cuadradas. Esto representa el área de la región R.

Frequently asked questions (FAQs)
What is the probability of rolling a fair dice and getting an even number?
+
What are the characteristics of the hyperbola function described by the equation (x^2/a^2) - (y^2/b^2) = 1?
+
What is the vertex of the quadratic function represented by the equation y = -2x^2 + 4x + 3?
+
New questions in Mathematics
reduction method 2x-y=13 x+y=-1
-11+29-18
90 divided by 40
For a temperature range between 177 degrees Celsius to 213 degrees Celsius, what is the temperature range in degrees Fahrenheit.
Express the following numbers in decimal system, where the subscript indicates the base: 110101 (SUBINDEX=2)
A National Solidarity Bond offers A 5 year bond offering a gross return of 15% Calculate the AER for this investment. (Give your answer to two decimal places, no need for the percent or € sign in your answer)
According to a survey in a country 27% of adults do not own a credit card suppose a simple random sample of 800 adults is obtained . Describe the sampling distribution of P hat , the sample proportion of adults who do not own a credit card
There were no defectives in a sample of 1 light bulb does this sample provide sufficient evidence that in the warehouse with millions of light bulbs fewer than 10% are defective?
What is 28 marks out of 56 as a percentage
v Is the following statement a biconditional? If Shannon is watching a Tigers game, then it is on television.
Calculate the boiling temperature and freezing temperature at 1 atmosphere pressure of a solution formed by dissolving 123 grams of ferrous oxide in 1.890 grams of HCl.
19) If the temperature of -8°C decreases by 12°C, how much will it be? a)-20°C -4°C c) 4°C d) 20°C
The points (-5,-4) and (3,6) are the ends of the diameter of the circle calculate subequation
2.380× (1+0.05) / 0.95−0.05
A company has had the following data for two consecutive years. Total, asset item 3,100,500 euros 3,300,550 euros. Net amount of business figures 4,755,250 euros /5,100 euros Average number of workers employed during the year 64/70 You can present a balance sheet in an abbreviated form
A diamond ring was reduced from $999.99 to $689.99. Find the percent reduction in the price. Round the answer to the nearest tenth of a percent, if necessary.
Write an expression using compatible numbers that can be used to estimate the quotient 629\86
Given a circle 𝑘(𝑆; 𝑟 = 4 𝑐𝑚) and a line |𝐴𝐵| = 2 𝑐𝑚. Determine and construct the set of all centers of circles that touch circle 𝑘 and have radius 𝑟 = |𝐴𝐵|
What js the greatest 4-digit even number that can be formed by 3,6,1,4?
calculate the product of 4 and 1/8