MathMaster logo

MathMaster

  1. Home
  2. general
  3. the concept of the integral arose from the need to calculate the area of a curved region for example calculating the area
Question

The concept of the integral arose from the need to calculate the area of a curved region. For example, calculating the area delimited by the region above the x-axis and below the graph of the function f(x) = x3 in a given limited interval is not such a simple task if we consider the knowledge acquired throughout basic education. But the definition of Definite Integral allows us to obtain this area precisely. Thus, given the function f(x) = x3, the area of the region below the graph of f, above the x-axis in the interval from 0 to 2 is equal to.

134

likes
672 views

Answer to a math question The concept of the integral arose from the need to calculate the area of a curved region. For example, calculating the area delimited by the region above the x-axis and below the graph of the function f(x) = x3 in a given limited interval is not such a simple task if we consider the knowledge acquired throughout basic education. But the definition of Definite Integral allows us to obtain this area precisely. Thus, given the function f(x) = x3, the area of the region below the graph of f, above the x-axis in the interval from 0 to 2 is equal to.

Expert avatar
Brice
4.8
113 Answers
1. \ \text{Evaluate the definite integral of the function:}
A = \int_{0}^{2} x^3 \, dx

2. \ \text{Find the antiderivative of} \ x^3:
\int x^3 \, dx = \frac{x^4}{4}

3. \ \text{Apply the limits of integration} \ (0 \ \text{to} \ 2):
A = \left[ \frac{x^4}{4} \right]_{0}^{2}

4. \ \text{Evaluate the antiderivative at the upper limit (2) and lower limit (0):}
A = \frac{2^4}{4} - \frac{0^4}{4}

5. \ \text{Simplify the expression:}
A = \frac{16}{4} - 0

6. \ \text{Finish the calculation:}
A = 4

\text{Therefore, the area is} \ 4

Frequently asked questions (FAQs)
What is the value of the third side in a right triangle, given that the adjacent side is 5 units long and the angle of inclination is 45 degrees?
+
What is the result of ((10 + 5) x 3) - 7 ?
+
What is the radian measure of an angle that subtends an arc length of 4π in a circle with radius 2?
+
New questions in Mathematics
How to find the value of x and y which satisfy both equations x-2y=24 and 8x-y=117
Let f(x)=||x|−6|+|15−|x|| . Then f(6)+f(15) is equal to:
𝑦 = ( 𝑥2 − 3) (𝑥3 + 2 𝑥 + 1)
a ferry travels 1/6 of the distance between two ports in 3/7 hour. the ferry travels at a constant rate. at this rate, what fraction of the distance between the two ports can the ferry travel in one hour?
431414-1*(11111-1)-4*(5*3)
-11+29-18
Write 32/25 as a percent
solve the following trigo equation for 0°<= x <= 360°. sec x =-2
2.5 / 21.85
An electrical company manufactures batteries that have a duration that is distributed approximately normally, with a mean of 700 hours and a standard deviation of 40 hours. Find the probability that a randomly selected battery has an average life of less than 810 hours.
2x+4x=
Primes are numbers divisible only by 1 and themselves; There are infinitely many prime numbers and the first ones are 2, 3, 5, 7, 11, 13, 17, 19, 23, .... Consider a 12-sided die, with the faces numbered from 1 to 12. Out of 4 rolls, the probability that only the first three numbers are primes is:
Raúl, Gilberto and Arturo are playing golf; The probabilities of winning for each one are as follows: (Raúl wins) = 20% (Gilberto wins) = 0.05% (Arturo wins) = ¾%. Perform operations and order events from least to most probable.
A machine produces 255 bolts in 24 minutes. At the same rate, how many bolts would be produced in 40 minutes?
prove that for sets SS, AA, BB, and CC, where AA, BB, and CC are subsets of SS, the following equality holds: (A−B)−C=(A−C)−(B−C)
Write the inequality in the form of a<x<b. |x| < c^2
8. Measurement Jillian measured the distance around a small fish pond to be 27 yards. What would be a good estimate of the distance across the pond: 14 yards, 9 yards, or 7 yards? Explain how you decided.
4m - 3t + 7 = 16
6(k-7) -2=5
To apply a diagnostic test, in how many ways can 14 students be chosen out of 25? if the order does not matter

You might be interested in