being, containing, or relating to one or more mathematical integers. (2) : relating to or concerned with mathematical integration or the results of mathematical integration. : formed as a unit with another part. a seat with integral headrest.
The definite integral has a unique value. A definite integral is denoted by ∫ab f(x) dx, where a is called the lower limit of the integral and b is called the upper limit of the integral.
Step-by-step explanation:
Definite Integral Definition
The definite integral of a real-valued function f(x) with respect to a real variable x on an interval [a, b]
Definite Integral as Limit of Sum
The definite integral of any function can be expressed either as the limit of a sum or if there exists an antiderivative F for the interval [a, b], then the definite integral of the function is the difference of the values at points a and b. Let us discuss definite integrals as a limit of a sum. Consider a continuous function f in x defined in the closed interval [a, b]. Assuming that f(x) > 0, the following graph depicts f in x.
The integral of f(x) is the area of the region bounded by the curve y = f(x). This area is represented by the region ABCD as shown in the above figure. This entire region lying between [a, b] is divided into n equal subintervals given by [x0, x1], [x1, x2], …… [xr-1, xr], [xn-1, xn].
Let us consider the width of each subinterval as h such that h → 0, x0 = a, x1 = a + h, x2 = a + 2h,…..,xr = a + rh, xn = b = a + nh
and n = (b – a)/h
Also, n→∞ in the above representation.
Now, from the above figure, we write the areas of particular regions and intervals as:
Area of rectangle PQFR < area of the region PQSRP < area of rectangle PQSE ….(1)
Answers & Comments
Verified answer
Step-by-step explanation:
being, containing, or relating to one or more mathematical integers. (2) : relating to or concerned with mathematical integration or the results of mathematical integration. : formed as a unit with another part. a seat with integral headrest.
Answer:
The definite integral has a unique value. A definite integral is denoted by ∫ab f(x) dx, where a is called the lower limit of the integral and b is called the upper limit of the integral.
Step-by-step explanation:
Definite Integral Definition
The definite integral of a real-valued function f(x) with respect to a real variable x on an interval [a, b]
Definite Integral as Limit of Sum
The definite integral of any function can be expressed either as the limit of a sum or if there exists an antiderivative F for the interval [a, b], then the definite integral of the function is the difference of the values at points a and b. Let us discuss definite integrals as a limit of a sum. Consider a continuous function f in x defined in the closed interval [a, b]. Assuming that f(x) > 0, the following graph depicts f in x.
The integral of f(x) is the area of the region bounded by the curve y = f(x). This area is represented by the region ABCD as shown in the above figure. This entire region lying between [a, b] is divided into n equal subintervals given by [x0, x1], [x1, x2], …… [xr-1, xr], [xn-1, xn].
Let us consider the width of each subinterval as h such that h → 0, x0 = a, x1 = a + h, x2 = a + 2h,…..,xr = a + rh, xn = b = a + nh
and n = (b – a)/h
Also, n→∞ in the above representation.
Now, from the above figure, we write the areas of particular regions and intervals as:
Area of rectangle PQFR < area of the region PQSRP < area of rectangle PQSE ….(1)