∫ calculus

1. **Stel het probleem vast:** We hebben de functie $$f(x) = 4 + \sqrt{9 - x^2}$$ en het vlakdeel $$V$$ begrensd door de lijn $$y=4$$, de y-as ($$x=0$$) en de grafiek van $$f$$ voo

1. **Stel het probleem vast:** Bereken algebraïsch de inhoud van het gebied $V$ dat ontstaat door de grafiek van een functie om de $y$-as te draaien. 2. **Formule voor inhoud om de

1. **State the problem:** Evaluate the definite integral $$\int_1^4 \frac{2}{\sqrt{x}} \, dx$$. 2. **Recall the formula and rules:** The integral of $$x^n$$ with respect to $$x$$ i

1. **State the problem:** Find the limit $$\lim_{x \to 3} \frac{\sqrt{1+x} - 2}{x - 3}$$. 2. **Recall the formula and rules:** This is a limit of the form $$\frac{f(x) - f(a)}{x -

1. The problem asks to determine and graph the domain of the function $$f(x,y) = \ln(x + y)$$. 2. The natural logarithm function $$\ln(z)$$ is defined only for $$z > 0$$.

1. **State the problem:** Calculate the first derivative of the function $$f(x) = \frac{x^2 - x}{x^2}$$. 2. **Rewrite the function:** Simplify the function before differentiating.

1. **Problem:** Given the function $$f(x) = xe^x$$, find the domain and asymptotes. 2. **Domain:** The function $$f(x) = xe^x$$ is defined for all real numbers because both $$x$$ a

1. **State the problem:** We need to plot the function $f(x) = x e^{-x}$ and identify key points: the origin $(0,0)$, the local maximum at $(1, \frac{1}{e})$, and the inflection po

1. **State the problem:** Find the limit \( \lim_{x \to -1} \frac{x^2 + x}{(x+1)^4} \). 2. **Recall the formula and rules:** When evaluating limits involving rational functions, if

1. **Problem statement:** Find the limit $$\lim_{x \to +\infty} \frac{\sqrt{-4x^2 + 2x + 1}}{1 - 9x^2}$$ and identify the horizontal asymptote. 2. **Formula and rules:** For limits

1. **Problem:** Do an example of an integral. 2. **Formula used:** For a basic power integral, use

1. **Stating the problem:** Find the derivative of the function $$f(x) = 2x^3 - 9x^2 + 12x$$ and then evaluate it at $$x = 2$$ to find $$f'(2)$$.

1. **Stating the problem:** We are given the equation $x^5 = 2x^3 + c$ and a point $(3,5)$ on the curve. We need to find the constant $c$ and verify the gradient (derivative) at $x

1. **Problem 3.1:** Find the points of intersection in the first quadrant of the polar curves $r_1 = \sin \theta$ and $r_2 = \sin(2\theta)$.\n\n2. The points of intersection satisf

1. **State the problem:** We need to find the indefinite integral $$\int 18x^2 (x^3+2)^5 \, dx.$$\n\n2. **Identify the method:** This integral suggests using substitution because t

1. The problem is to find the extrema of the function $$y = (x - 2)^4 - 1$$ using higher order derivatives. 2. To find extrema, we first find the first derivative $$y'$$ and set it

1. We are asked to evaluate the definite integral $$\int_0^1 \frac{x^2}{\sqrt{x^3 + 1}} \, dx$$. 2. To solve this integral, we use substitution. Let $$u = x^3 + 1$$. Then, $$\frac{

1. We are asked to evaluate the integral $$\int \cos^7\left(\frac{2x}{5}\right) \cdot \sin^6\left(\frac{2x}{5}\right) \, dx.$$\n\n2. The integral involves powers of sine and cosine

1. **State the problem:** Find the length of the curve given by the vector function $$\mathbf{r}(t) = (\sqrt{2}t, e^t, e^{-t})$$ for $$0 \leq t \leq 1$$. 2. **Formula for curve len

1. **State the problem:** We are given the vector function $$r(t) = \left(\tan(t) \cot(t), \ln t, \sqrt{\pi^2 - t^2}\right)$$ and need to find its domain and the limit $$\lim_{t \t

1. **State the problem:** Find the area bounded by the curves $y = x^2$ and $y = x$. 2. **Identify the points of intersection:** Set $x^2 = x$ to find where the curves intersect.