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 result of dividing 3/4 by 2/3?
+
Question: Find the basis of vectors in R³ formed by the columns of matrix A = [2 -1 3; 0 1 -4; 1 0 5].
+
What is the limit of the function (5x^3 - 2x^2 + 7)/(2x^3 + 3x^2 + 5) as x approaches 2?
+
New questions in Mathematics
Find 2 numbers that the sum of 1/3 of the first plus 1/5 of the second will be equal to 13 and that if you multiply the first by 5 and the second by 7 you get 247 as the sum of the two products with replacement solution
2x-y=5 x-y=4
4x-3y=5;x+2y=4
Margin of error E=0.30 populations standard deviation =2.5. Population means with 95% confidence. What I the required sample size (round up to the whole number)
"If three wolves catch three rabbits in three hours, how many wolves would it take to catch a hundred rabbits in a hundred hours?" The answer is the number of response units.
A study reports the following final notation: F (3, 32) = 9.50, p < .05. How many total participants were involved in this study? Group of answer choices 34 32 36
Suppose you have a sample of 100 values from a population with mean mu = 500 and standard deviation sigma = 80. Given that P(z < −1.25) = 0.10565 and P(z < 1.25) = 0.89435, the probability that the sample mean is in the interval (490, 510) is: A)78.87% B)89.44% C)10.57% D)68.27%
Determine the reduced equation of the straight line that is perpendicular to the straight line r: y=4x-10 and passes through the origin of the Cartesian plane
Determine the increase of the function y=4x−5 when the argument changes from x1=2 to x2=3
The sick-leave time of employees in a firm in a month is normally with a mean of 100 hours and a standard deviation of 20 hours. Find the probability that the sick-leave time of an employee in a month exceeds 130 hours.
Quadratic equation 2X = 15/X + 7
A function is considered exponential when it has a base with positive values greater than zero and different from one, where the exponent is an unknown. An important characteristic of exponential functions is that they show rapid growth or decay as an independent variable increases or decreases. Given the function 25^(x+3)=125, it is calculated that x has the value of
Find the center coordinates and radius of a circle for an equation written as: 3x2 + 3y2 - 6y = —12× + 24
Solve for B write your answer as a fraction or as a whole number. B-1/7=4
Arturo had hospitalization expenses of $8,300. Your policy for medical expenses Seniors have a deductible of $500 and expenses are paid at a 20% coinsurance. These are the first expenses ever this year, how much will Arturo have to pay in your bill for hospitalization expenses?
9n + 7(-8 + 4k) use k=2 and n=3
How many digits are there in Hindu-Arabic form of numeral 26 × 1011
5a-3.(a-7)=-3
12[4 + (8 + 7) + 5]
5 1/9 + 2 2/3