Infi-2 7

Continuity of area function #lemma

\begin{matrix} Let f be Riemann-integrable on [a, b] \\ Then S (x) = \int_{a}^{x} f (t) d t is continuous on [a, b] \end{matrix}

\begin{matrix} lim_{x \to c} S (x) = S (c) \end{matrix}

\begin{matrix} lim_{x \to 0} \frac{\int_{0}^{x} \sin (t^{2}) d t}{9 x^{3}} \overset{L}{=} lim_{x \to 0} \frac{\sin (x^{2})}{27 x^{2}} = \frac{1}{27} \end{matrix}

\begin{matrix} \int_{g (x)}^{h (x)} f (t) d t = \int_{0}^{h (x)} f (t) d t - \int_{0}^{g (x)} f (t) d t = S (h (x)) - S (g (x)) \\ ⟹ {(\int_{g (x)}^{h (x)} f (t) d t)}^{'} = (S (h (x)) - S (g (x)))^{'} = h^{'} (x) f (h (x)) - g^{'} (x) f (g (x)) \end{matrix}

\begin{matrix} {(\int_{\sqrt{x}}^{x^{2}} \sin (t^{2}) d t)}^{'} = {(\int_{0}^{x^{2}} \sin (t^{2}) d t - \int_{0}^{\sqrt{x}} \sin (t^{2}) d t)}^{'} = \sin (x^{4}) \cdot 2 x - \sin x \cdot \frac{1}{2 \sqrt{x}} \end{matrix}

\begin{matrix} Let F be a primitive of f \\ \int_{a}^{b} f (x) d x = F (b) - F (a) \end{matrix}

\begin{matrix} 1. & Calculating area \\ area between graphs of functions f, g on [a, b] \\ is equal to \int_{a}^{b} | f (x) - g (x) | d x \\ 2. & Calculating volume of a revolution (Pappus theorem) \\ volume of a revolution is equal to \\ {\begin{matrix} V_{X} (f) = π \int_{a}^{b} f^{2} (x) d x when rotating around axis X \\ V_{Y} (f) = 2 π \int_{a}^{b} x f (x) d x when rotating around axis Y \end{matrix} \\ 3. & Arc length \\ length of the arc of continuously differentiable function on [a, b] \\ is equal to L (f) = \int_{a}^{b} \sqrt{1 + (f^{'} (x))^{2}} d x \\ 4. & Revolution surface area \\ Revolution surface area of continuously differentiable function on [a, b] \\ is equal to A (f) = 2 π \int_{a}^{b} f (x) \sqrt{1 + (f^{'} (x))^{2}} d x \end{matrix}

\begin{matrix} Proof for 3. \\ Let P = {x_{0}, \dots, x_{n}} \\ L (f) \approx \sum_{i = 1}^{n} \sqrt{(x_{i} - x_{i - 1})^{2} + (f (x_{i}) - f (x_{i - 1}))^{2}} \\ Let Δ x_{i} = x_{i} - x_{i - 1} \\ Let Δ f (x_{i}) = f (x_{i}) - f (x_{i - 1}) \\ ⟹ L (f) \approx \sum_{i = 1}^{n} Δ x_{i} \sqrt{1 + {(\frac{Δ f (x_{i})}{Δ x_{i}})}^{2}} \\ By the Mean value theorem: \exists c_{i} \in [x_{i - 1}, x_{i}] : f^{'} (c_{i}) = \frac{Δ f (x_{i})}{Δ x_{i}} \\ Let C = {c_{1}, \dots, c_{n}} \\ ⟹ \sum_{i = 1}^{n} \sqrt{1 + f^{'} (c_{i})^{2}} Δ x_{i} = S (f, P, C) \\ ⟹ L (f) = lim_{λ (P) \to 0} S (f, P, C) = \int_{a}^{b} \sqrt{1 + (f^{'} (x))^{2}} d x \end{matrix}

\begin{matrix} \sinh x = \frac{e^{x} - e^{- x}}{2} \\ \cosh x = \frac{e^{x} + e^{- x}}{2} \\ Let us calculate the arc length of hyperbolic cosine on [0, 1] \\ (\cosh x)^{'} = \frac{e^{x} - e^{- x}}{2} = \sinh x \\ L (\cosh) = \int_{0}^{1} \sqrt{1 + {(\frac{e^{x} - e^{- x}}{2})}^{2}} d x \\ \sqrt{1 + {(\frac{e^{x} - e^{- x}}{2})}^{2}} = \sqrt{\frac{4 + e^{2 x} - 2 + e^{- 2 x}}{4}} = \frac{\sqrt{e^{2 x} + 2 + e^{- 2 x}}}{2} = \frac{e^{x} + e^{- x}}{2} = \cosh x \\ ⟹ L (\cosh) = \int_{0}^{1} \cosh x d x = \int_{0}^{1} \frac{e^{x} + e^{- x}}{2} d x = {(\frac{e^{x} - e^{- x}}{2})}_{x = 0}^{x = 1} = \frac{e - e^{- 1}}{2} \end{matrix}