∫ calculus

1. **State the problem:** We need to find the total area of the shaded regions enclosed by the curve $$y = 12x^3 - 48x^2 + 36x$$ and the x-axis, between the roots $$x=0$$, $$x=1$$,

1. Problemet är att bestämma konstanten $a$ så att $$\int_0^2 f(x) \, dx = f(a)$$ där $$f(x) = \frac{\sqrt{2x}}{x}.$$\n\n2. Vi börjar med att förenkla funktionen $f(x)$. Eftersom $

1. **State the problem:** Evaluate the integral $$\frac{1}{2} \int_{-1}^{1} \sqrt{1 - x^2} \, dx$$. 2. **Recognize the integral:** The integral $$\int_{-1}^{1} \sqrt{1 - x^2} \, dx

1. **State the problem:** We are given the degree 4 Taylor polynomial \(P_4(x) = 9 + \frac{1}{7}(x-2)^3 + 7(x-2)^4\) for a function \(f(x)\) centered at \(x=2\). We want to find an

1. **State the problem:** Evaluate the double integral

1. **Enunciado do problema:** Calcule o volume da região abaixo do gráfico da função $$f(x,y) = \begin{cases} \frac{x^2 - y^2}{\sqrt{x^2 + y^2}} & \text{se } (x,y) \neq (0,0) \\ 0

1. State the problem: Find $$\lim_{x\to 3}\frac{x^2-9}{x-3}.$$\n\n2. Substitute directly (quick check): The form is $$\frac{0}{0}$$ because $$x^2-9=(x-3)(x+3).$$\n\n3. Factor the n

1. **State the problem:** Find the limit $$\lim_{x \to -3} \frac{ \sqrt{2}\sqrt{x^2 - x} - \sqrt{6}\sqrt{x^2 + x - 6} }{ x^2 - 4x + 3 }$$

1. **Problem statement:** We need to find the least positive integer $m$ such that the temperature $P(m)$ of the ingot is below 60°C. 2. **Given function:**

1. **State the problem:** We have the function $f(x) = -\frac{1}{3}x^3 + 2x^2 - 12$.

1. **Problem statement:** Given the function $f(x) = -\frac{1}{3}x^3 + 2x^2 - 12$, find: i) The intervals where $f$ is increasing or decreasing and the local maxima and minima.

1. Problem: Evaluate $$\lim_{x\to 3}\,\frac{x^2-9}{x-3}.$$\n

1. State the problem: Evaluate the limit $\lim_{x\to 2} \frac{x^2-4}{x-2}$. 2. Identify the type of limit: Plugging in $x=2$ gives an indeterminate form $\frac{0}{0}$.

1. **Problem statement:** Given the function $f(x) = -\frac{1}{3}x^3 + 2x^2 - 12$, find: i) The intervals where $f$ is increasing or decreasing and the local maxima and minima.

1. **State the problem:** We need to find the derivative of the function $$f(x) = \frac{x}{x+1}$$ using the definition of the derivative. 2. **Recall the definition of the derivati

1. **State the problem:** We need to find the indefinite integral of the function $3x + e^x$ with respect to $x$. 2. **Formula used:** The integral of a sum is the sum of the integ

1. **State the problem:** Evaluate the integral $$\int 6 \cdot 2^{3x} \, dx$$. 2. **Recall the formula:** The integral of an exponential function $$a^{kx}$$ with respect to $$x$$ i

1. **State the problem:** We need to find the derivative of the function $$f(x) = \frac{x}{x+1}$$ using the definition of the derivative. 2. **Recall the definition of the derivati

1. **State the problem:** We need to find the derivative of the function $$f(x) = x^{x+1}$$ using the definition of the derivative. 2. **Recall the definition of the derivative:**

1. **State the problem:** We need to find the integral of the function $5e^{2x}$ with respect to $x$. 2. **Recall the formula:** The integral of $e^{ax}$ with respect to $x$ is giv

1. **State the problem:** Evaluate the integral $\int \cos(2x - 4x) \, dx$. 2. **Simplify the argument of cosine:**