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 measure of an angle bisector if the adjacent angles formed are 62° and 118°?
+
What is the cube root function's graph behavior in terms of concavity, symmetry, and x-intercept?
+
What is the value of the tangent of angle A if the opposite side length is 5 and the adjacent side length is 3?
+
New questions in Mathematics
2x-y=5 x-y=4
Calculate the equation of the tangent line ay=sin(x) cos⁡(x)en x=π/2
If O(3,-2) is reflected across x = 2. What are the coordinates of O
7273736363-8
(5y 9)-(y 7)
Three squares have a total area of 35.25 𝑐𝑚2 . The larger square has twice the side-length of the middle-sized square. The smaller square has its side length exactly 0.5 cm smaller than the middle-sixed square. Find the side lengths of each of the three squares.
The physician orders 15mg of tramadol(liquid). On hand is 30mg/2mL vials. How many mL will the MA administer?
4+168×10³×d1+36×10³×d2=-12 -10+36×10³×d1+72×10³×d2=0
Quadratic equation 2X = 15/X + 7
If the regression equation is given by 4x –y + 5 = 0, then the slope of regression line of y on x is
A car travels 211 miles on 15 gallons of gasoline. The best estimate of the car’s miles per gallon is?
Determine a general formula​ (or formulas) for the solution to the following equation.​ Then, determine the specific solutions​ (if any) on the interval [0,2π). cos30=0
A contractor gives a bank note for $10250 at a rate of 1% for one month. How much interest is charged for 4 months?
16.What payment (deposit) made at the end of each month will accumulate to $10473 in 13 years at 7.9% compounded monthly? Enter to the nearest cent (two decimals). Do not use $ signs or commas in the answer.
Log0
A given initial capital in simple interest at the annual rate and for 27 months produced the accumulated capital of 6600 um if the same capital had been invested at the same rate but during 28 months the said accumulated capital would be increased in an amount corresponding to 0.75% of the initial capital Calculate the initial capital and the annual rate at which it was invested
The area bounded by the curve y=ln(x) and the lines x=1 and x=4 above the x−axis is
The average weekly earnings in the leisure and hospitality industry group for a re‐ cent year was $273. A random sample of 40 workers showed weekly average ear‐ nings of $285 with the population standard deviation equal to 58. At the 0.05 level of significance can it be concluded that the mean differs from $273? Find a 95% con‐ fidence interval for the weekly earnings and show that it supports the results of the hypothesis test.
Carmen's age was twice as old as Luis was when Carmen was Luis's age. When Luis is Carmen's age, their ages will add up to 112.
5 1/9 + 2 2/3

You might be interested in