# How to Reshape Plastic Sunglasses

## 20.2 Relationship between integral and differential calculus

What does the 1st law of integral and differential calculus say? If a function f is integrable on [a, b], then F (x) = \ (\ int_a ^ xf (t) dt \) and the following applies: F '(x) = f (x), if f (x ) continuous in [a, b] Functions that are continuous on an interval [a, b] are also there ..? integrable What is the so-called "sub-integral"? The supremum of all sub-sums: sup {U (f, P) | P partition of [a, b]} What is the so-called "upper integral"? The infimum of all upper sums: inf {O (f, P) | P partition of [a, b]} What does the second law of the differential and integral calculus say? Is f integrable on [a, b] and g an antiderivative of f: ==> \ (\ int_a ^ b f (x) dx = g (b) - g (a) \) is g '(x) = f and \ (\ int_a ^ b f (x) dx \) = g (b) - g (a) How can one then write g (b) -g (a)? As: \ ([g (x)] _ a ^ b \) or \ (g (x) | _a ^ b \) How does partial integration work? one analyzes whether the function can be represented as the product of a function and a derivative f (x) g '(x). Then one forms the difference: \ (f (x) g (x) | _a ^ b \) and subtracts the integral with "exchanged roles", i.e. from f '(x) g (x) What is the formula for partial integration? \ (\ int_a ^ b f (x) g '(x) dx = \) \ (f (x) g (x) | _a ^ b - \ int_a ^ b f' (x) g (x) dx \) Where can the rule for partial integration be derived from? By transformation from the product rule of differentiation Where can the substitution rule of integration be derived from? From the chain rule of integration What is the substitution rule? \ (\ int_a ^ b f (g (x)) g '(x) dx = \ int_ {g (a)} ^ {g (b)} f (u) du \) When can the substitution rule of integration be used? In the case of chained functions to which a factor is "attached", which is the derivative of the internal function. If one writes g '(x) as dg (x), how can the substitution rule be represented? What's the advantage of that? than: \ (\ int_a ^ bf (g (x)) dg (x) = \ int_ {g (a)} ^ {g (b)} f (u) you \) is easier to remember ..: simply g Replace (x) with u, a with g (a) and b with g (b) What is the area of the unit circle? How can this be proven by integrals? Pi, by converting the circular formula \ (x ^ 2 + y ^ 2 = 1 \) into a function that describes a semicircle above the x-axis: \ (\ sqrt {(1-x ^ 2)} \) and then calculates the area through integration

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