Notes On Fundamental Theorem of Calculus - CBSE Class 12 Maths
The area of the region bounded by y = f(x), the ordinates x = a, x =b, and the X-axis is ∫ab f(x) dx, where a ≤ x ≤ b. Let x be any point in the closed interval [a, b] such that f(x) > 0 ∀ x ∈ [a,b]. ∫ar f(x) dx represents the area of the shaded region. The area of the shaded region depends on the value of x. Therefore, the area of the shaded region is a function of x. We represent the function of x as A(x). Area function , A(x) = ∫ax f(x) dx First fundamental theorem on integral calculus: 1) If f is a continuous function defined on Closed interval [a, b] and A(x) be the area function, then A'(x) = f(x), ∀ x ∈ [a,b]. Second fundamental theorem on integral calculus: 2) Let f be a function defined on closed interval [a, b] and let g be an anti-derivate of function f, then: ∫ab f(x) dx = (Value of the anti- derivative of g of f at the upper limit) - (value of the anti - derivative g o f at lower limit) Mathematically, it can be denoted as ∫ab f(x) dx = [g(x)]ab = g(b) - g(a) This theorem is applicable only when f is well defined in the closed interval [a, b]. Steps to calculate ∫ab f(x) dx: Step: 1) Find ∫ f(x) dx Let g(x) = ∫ f(x) dx. Step: 2) ∫ab f(x) dx = [g(x) + C]ab  = [g(b) + C] - [g(a) + C] = g(b) - g(a)

#### Summary

The area of the region bounded by y = f(x), the ordinates x = a, x =b, and the X-axis is ∫ab f(x) dx, where a ≤ x ≤ b. Let x be any point in the closed interval [a, b] such that f(x) > 0 ∀ x ∈ [a,b]. ∫ar f(x) dx represents the area of the shaded region. The area of the shaded region depends on the value of x. Therefore, the area of the shaded region is a function of x. We represent the function of x as A(x). Area function , A(x) = ∫ax f(x) dx First fundamental theorem on integral calculus: 1) If f is a continuous function defined on Closed interval [a, b] and A(x) be the area function, then A'(x) = f(x), ∀ x ∈ [a,b]. Second fundamental theorem on integral calculus: 2) Let f be a function defined on closed interval [a, b] and let g be an anti-derivate of function f, then: ∫ab f(x) dx = (Value of the anti- derivative of g of f at the upper limit) - (value of the anti - derivative g o f at lower limit) Mathematically, it can be denoted as ∫ab f(x) dx = [g(x)]ab = g(b) - g(a) This theorem is applicable only when f is well defined in the closed interval [a, b]. Steps to calculate ∫ab f(x) dx: Step: 1) Find ∫ f(x) dx Let g(x) = ∫ f(x) dx. Step: 2) ∫ab f(x) dx = [g(x) + C]ab  = [g(b) + C] - [g(a) + C] = g(b) - g(a)

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