Question

calculates the exact area under the curve using Riemann sums of f(x)= 4-x2 on an interval [ 1,2]

251

likes
1254 views

Answer to a math question calculates the exact area under the curve using Riemann sums of f(x)= 4-x2 on an interval [ 1,2]

Expert avatar
Dexter
4.7
109 Answers
Para calcular el área exacta bajo la curva mediante sumas de Riemann de la función f(x) = 4 - x^2 en el intervalo [1,2], vamos a seguir los siguientes pasos:

Paso 1: Dividir el intervalo [1,2] en subintervalos iguales. Vamos a elegir n subintervalos para aproximar el área.

Paso 2: Calcular el ancho de cada subintervalo. Encuentra el valor de \Delta x dividiendo la longitud del intervalo (2 - 1) entre n, es decir, \Delta x = \frac{{2 - 1}}{n}.

Paso 3: Determinar los puntos de evaluación. Escoge un punto dentro de cada subintervalo para evaluar la función. Aquí utilizaremos el punto medio de cada subintervalo.

Paso 4: Calcular el área aproximada bajo la curva. Para cada subintervalo, calculamos el área de un rectángulo cuya altura es el valor de la función evaluada en el punto medio del subintervalo, y cuya base es el ancho del subintervalo. Luego, sumamos todas estas áreas de los rectángulos para obtener una aproximación del área bajo la curva.

Paso 5: Tomar el límite cuando n tiende a infinito. A medida que aumentamos el número de subintervalos, la aproximación del área se vuelve más precisa. Tomando el límite cuando n tiende a infinito, obtendremos el área exacta bajo la curva.

Ahora vamos a calcular el área exacta utilizando los métodos de sumas de Riemann:

Paso 1: Dividir el intervalo [1,2] en subintervalos iguales. Tomaremos n subintervalos.

Paso 2: Calcular el ancho de cada subintervalo. Tenemos \Delta x = \frac{{2 - 1}}{n} = \frac{1}{n}.

Paso 3: Determinar los puntos de evaluación. Utilizaremos el punto medio de cada subintervalo.

Paso 4: Calcular el área aproximada bajo la curva. Para cada subintervalo i, el punto de evaluación será x_i^* = 1 + \frac{\Delta x}{2} + i \cdot \Delta x, y el área del rectángulo correspondiente será A_i = f(x_i^*) \cdot \Delta x. Entonces, el área aproximada A será la suma de todas estas áreas:

A = \sum_{i=0}^{n-1} A_i = \sum_{i=0}^{n-1} f\left(1 + \frac{\Delta x}{2} + i \cdot \Delta x\right) \cdot \Delta x

El límite de esta suma cuando n tiende a infinito nos dará el área exacta bajo la curva.

Paso 5: Tomar el límite cuando n tiende a infinito. Es decir:

\lim_{{n \to \infty}} \sum_{i=0}^{n-1} f\left(1 + \frac{\Delta x}{2} + i \cdot \Delta x\right) \cdot \Delta x

Para calcular el límite de esta suma, podemos utilizar el teorema fundamental del cálculo o notar que la función f(x) = 4 - x^2 es continua en el intervalo [1,2] y, por lo tanto, integrable. Por lo tanto, el área exacta bajo la curva se puede calcular mediante la integral definida de la función en el intervalo [1,2]:

A = \int_{1}^{2} (4 - x^2) \, dx

Ahora podemos proceder a calcular la integral para obtener el área exacta:

\int_{1}^{2} (4 - x^2) \, dx = \left[ 4x - \frac{x^3}{3} \right]_{1}^{2}

Evaluamos la integral en los límites de integración:

= \left[ 4(2) - \frac{(2)^3}{3} \right] - \left[ 4(1) - \frac{(1)^3}{3} \right]

= \left[ 8 - \frac{8}{3} \right] - \left[ 4 - \frac{1}{3} \right]

= \left[ \frac{24}{3} - \frac{8}{3} \right] - \left[ \frac{12}{3} - \frac{1}{3} \right]

= \frac{16}{3} - \frac{11}{3}

= \frac{5}{3}

Entonces, el área exacta bajo la curva f(x) = 4 - x^2 en el intervalo [1,2] es \frac{5}{3}.

\textbf{Respuesta:} El área exacta bajo la curva mediante sumas de Riemann de f(x) = 4 - x^2 en el intervalo [1,2] es \frac{5}{3}.

Frequently asked questions (FAQs)
What are the characteristics of an ellipse with major axis length 8 and minor axis length 6?
+
What is the domain of the trigonometric function f(x) = cos(x) + sin(2x) over the interval [-π/4,π/4]?
+
What is the sum of vector A (-3, 8) and vector B (5, -6), after subtracting vector C (2, 4)?
+
New questions in Mathematics
The time it takes for a person to travel 300 m is 15 minutes. What is their speed in meters per second?
A college believes that 22% of applicants to that school have parents who have remarried. How large a sample is needed to estimate the true proportion of students who have parents who have remarried to within 5 percentage points?
The Lenovo company manufactures laptop computers, it is known that for every 60 laptops produced, 54 go on the market with the highest quality standards. If a sample of 15 laptops is taken, calculate the probability that: Exactly 2 are not of high quality
Additionally, the boss asked Armando to determine how many toy sales branches he would have in the fifteenth year, knowing that the first year they started with two branches, by the second they already had 5 branches and, by the third year, they had 8 branches. From the above, determine the number of branches it will have for the fifteenth year.
The bus one way of the road which is 10km is heading with speed of 20km/h ,then the bus the other 10km is heading with speed of 60km/h. The middle speed of the road is it equal with arithmetic speed of the v1 and v2 ?
Suppose X has a Poisson distribution, with a mean of 0.4. Determine the probability that x is at most 2.
What is 28 marks out of 56 as a percentage
The maximum gauge pressure of a hydraulic ramp is 16 atm, with a support area whose diameter is 20 cm. What is the mass of the heaviest vehicle that can be lifted?
From 1975 through 2020 the mean annual gain of the Dow Jones Industrial Average was 652. A random sample of 34 years is selected from this population. What is the probability that the mean gain for the sample was between 400 and 800? Assume the standard deviation is 1539
In a company dedicated to packaging beer in 750 mL containers, a normal distribution is handled in its packaging process, which registers an average of 745 mL and a standard deviation of 8 mL. Determine: a) The probability that a randomly selected container exceeds 765 mL of beer b) The probability that the beer content of a randomly selected container is between 735 and 755 mL.
What is 75 percent less than 60
X~N(2.6,1.44). find the P(X<3.1)
A popular cell phone family plan provides 1500 minutes. It charges 89.99/month for the first 2 lines and 9.99 for every line after that. Unlimited text messages for all phone lines costs $30.00/month, and Internet costs $10.00/month per phone line. If a family with a $200 monthly budget buys this plan and signs up for unlimited text messaging and Internet on each phone line, how many cell phone lines can they afford? Use an inequality to solve this problem. Graph your solution on the number line and explain the meaning of your graph in a sentence.
List the remaining zeros of the polynomial with the given zeros Zeros are: 2, 3i, and 3 + i
To find the increased amount on a standard term deposit with the following conditions: starting amount: BGN 13000, type of deposit: annual, annual compound interest rate: 1.4%, after 4 years;
The annual real property tax liability for a residential property is $4302 and has been paid by the seller in advance of closing. Using the 30-day month/260-day year method what will be the tax proration entry on the settlement statement round to the nearest dollar for a closing on Oct. 26 if the buyer owns the day of closing? a. $3525 credit to the buyer and $777 debit to the seller b. $777 debit to the buyer and $3525 debit to the seller c. $777 credit to the buyer and $777 debit to the seller d. $3525 debit to the buyer and $3525 credit to the seller *Can anyone help with this? I am studying for my real estate exam and am having trouble with some of the calculations :)
Total Users with an active Wise account = Total Active Users + Total Users who haven’t transacted Total Active Users = Total MCA Users + Total Send Users = Total New Users + Retained Users Total New Users = New Send Users + New MCA Users Total MCA Users = New MCA Users + Retained Users who transacted this month via MCA Total Send Users = New Send Users + Retained Users who transacted this month via Send Send CR = Total Send Users / Total Users with an active Wise account MCA CR = Total MCA Users / Total Users with an active Wise account New Send CR = New Send Users / New Profiles Created in Month New MCA CR = New MCA Users / New Profiles Created in Month We have recently witnessed a drop in MCA conversion, but send user conversion is stable, can you help explain why?
A property sold for $745,000 in a co-brokered transaction. The seller has agreed to pay a 7% commission to the listing firm. The listing firm has agreed to equally split the commission with the selling firm. If the buyer’s broker will receive 8% of the selling firm’s commission, how much commission will the buyer’s broker receive? $14,900 $3725 $$37250 $18625
If sin A=0.3 and cos A=0.6, determine the value of tan A.
5 1/9 + 2 2/3