∫ calculus

1. **State the problem:** We want to find the maximum value of the function $$y = 2x - x^3$$ and determine the maximum value of $$c$$ such that $$2x - x^3 = c$$ has solutions. 2. *

1. **Stating the problem:** Analyze the piecewise function defined as:

1. **Problem:** Find the area of the region bounded by the curves $y = x^2 + 2x$, $y = x + 6$, and the vertical lines $x = -4$ and $x = 4$. Set up the area as the sum of three inte

1. **State the problem:** We need to find the value of the definite integral $$\int_{-8}^{4} f(x) \, dx$$ where the function $f$ consists of three parts: a line segment from $(-8,5

1. **Problem:** Understand and apply basic integral formulas. 2. **Formula 1:** \(\int k \, dx = kx + c\), where \(k\) is a constant.

1. **Problem statement:** Differentiate the following functions: a) $g(x) = (x^2 - 2)(2x + 3)$

1. **Problem statement:** Given the graph of a function $f$, identify intervals where $f$ is increasing or decreasing, and find local and absolute extrema. 2. **Definitions and rul

1. **Problem statement:** Find the area of region T bounded by the y-axis. 2. **Given:** Region T is bounded by the y-axis and some curve (not explicitly given here, but assumed fr

1. **Problem statement:** We are given a function $$f(x) = -5 \cdot 10^{-5} \cdot x^4 + 0.06 \cdot x^2$$ and asked to compute two volumes: - $$V = \int_0^{60} f(x) \, dx$$

1. **Problem statement:** We have a sink profile defined by the function $$f(x) = -5 \cdot 10^{-5} x^4 + 0.06 x^2$$ over a horizontal span of 60 cm (from $x=0$ to $x=60$). We want

1. **Problem statement:** We have a sink with a profile described by the function $$f(x) = -5 \cdot 10^{-5} x^4 + 0.06 x^2$$ where $x$ is in centimeters and $f(x)$ gives the depth

1. Problemet är att beräkna integralen $$\int_1^2 3x^2 \, dx$$. 2. Formeln för integrering av en potensfunktion $$x^n$$ är $$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$$ där $$C$$ är

1. The problem is to find the area of the region bounded by the curves \(f(x) = x^3 + 2x^2 - x - 2\) and \(g(x) = 4x + 4\). 2. To find the area between two curves, we first find th

1. **State the problem:** Calculate the definite integral $$\int_2^8 x \, dx$$. 2. **Formula and rules:** The integral of $$x$$ with respect to $$x$$ is given by $$\int x \, dx = \

1. **State the problem:** Evaluate the expression $$1 + \int (1 - x^3) - (1 - x) \, dx$$.\n\n2. **Rewrite the integral:** The integral expression is ambiguous without limits, so we

1. We are given the function $f(x) = -2 \cos x$ and asked to find its derivative $g(x)$ and then evaluate $g\left(\frac{\pi}{2}\right)$. 2. The derivative of $\cos x$ is $-\sin x$,

1. **State the problem:** We have the function $f(x) = 2\sqrt{x}$ defined on the interval $[0,4]$.

1. **State the problem:** Find the area of the region $R$ bounded by the y-axis, the line $y=1$, and the curve $y=\sqrt[3]{x}$ by writing $x$ as a function of $y$ and integrating w

1. **Problem:** Find the derivative of the implicit function given by $$x^3 + y^2 = -x^5$$. 2. **Formula and rules:** Use implicit differentiation. Differentiate both sides with re

1. **Problem:** Find the derivative of $f(x) = e^{3x}$. 2. **Formula:** The derivative of $e^{u(x)}$ is $e^{u(x)} \cdot u'(x)$.

1. **Problem:** Calculate the limit $$\lim_{x \to 0} \frac{3e^x - 3}{2x}$$. 2. **Formula and rules:** This is a limit of the form $$\frac{f(x) - f(a)}{x - a}$$ as $$x \to a$$, whic