∫ calculus

1. **State the problem:** Find the first partial derivative of the function $$f(x,y) = e^{x+y} + y^2 \sin(x)$$ with respect to $$x$$. 2. **Recall the formula and rules:**

1. **State the problem:** Find the first partial derivative of the function $$f(x,y) = y^2 e^x + \tan(xy)$$ with respect to $$x$$. 2. **Recall the formula and rules:**

1. **State the problem:** We are given the derivative of a function $f'(x)$ and want to find the original function $f(x)$ and analyze its critical points. 2. **Given:**

1. **State the problem:** We are given the derivative $\frac{dy}{dx} = \sin\left(x + \frac{\pi}{3}\right)$ and the initial condition $y\left(\frac{\pi}{6}\right) = 3$. We need to f

1. **Stel het probleem vast:** We willen de primitieve (onbepaalde integraal) vinden van de functie $$f(x) = \frac{\arcsin(2x)}{\sqrt{1-4x^2}}$$. 2. **Formule en regels:** We gebru

1. The problem asks whether to use the washer or disk method for finding volumes of solids of revolution. 2. The disk method is used when the solid is formed by revolving a region

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