∫ calculus

1. **State the problem:** Calculate the value of $$\sqrt{\int_{3\pi/8}^{7\pi/8} 40^2 \, dx + \frac{\int_{7\pi/8}^{2\pi+3\pi/8} (-80)^2 \, dx}{2\pi}}$$.

1. نبدأ ببيان المشكلة: لدينا متوازي سطوح مستطيلة قاعدة مربعة الشكل، حجم المتوازي ثابت ويساوي 80 سم³. 2. المعطيات:

1. The problem asks: What does the first derivative represent in calculus? 2. The first derivative of a function $f(x)$, denoted as $f'(x)$ or $\frac{df}{dx}$, measures the rate at

1. **State the problem:** Find the derivative of the function $$f(x) = \sin x \tan x$$. 2. **Recall the product rule:** If $$f(x) = u(x)v(x)$$, then $$f'(x) = u'(x)v(x) + u(x)v'(x)

1. **State the problem:** Find the derivative of the function $$y = e^{\sqrt{7x^2 - 1}}$$. 2. **Recall the formula:** The derivative of $$e^{u(x)}$$ with respect to $$x$$ is $$\fra

1. Evaluate $$\int_0^1 x \sin(x^2) \, dx$$ - Use substitution: let $$u = x^2$$, then $$du = 2x \, dx$$ or $$x \, dx = \frac{du}{2}$$.

1. **Problem Statement:** Given that $\lim_{x \to a} f(x) = A$ and $\lim_{x \to a} g(x) = B$, we want to understand the rules for limits involving sums, products, quotients, and po

1. **State the problem:** We want to find the limit $$\lim_{x \to 1} \frac{x - 1}{x^2 - 1}$$. 2. **Recall the formula and rules:** The limit of a function as $x$ approaches a value

1. The problem asks to find the Stammfunktion (antiderivative) of the function $f(x) = \frac{3}{8}(x - 2)^2$. 2. Recall the power rule for integration: $$\int x^n \, dx = \frac{x^{

1. **State the problem:** We need to find $\frac{dy}{dx}$ given the parametric equations $x = t \sin t$ and $y = \cos t$ at $t = \frac{\pi}{2}$. 2. **Recall the formula:** For para

1. Το πρόβλημα ζητά να βρούμε τη σειρά Taylor της συνάρτησης $$f(x) = \frac{\sin x}{x^2 + 3x + 2}$$ μέχρι και τον τρίτο όρο (τρίτο παράγοντα).\n\n2. Η σειρά Taylor μιας συνάρτησης

1. Το πρόβλημα είναι να υπολογίσουμε τη δεύτερη παράγωγο μιας συνάρτησης. 2. Η δεύτερη παράγωγος είναι η παράγωγος της πρώτης παραγώγου, δηλαδή αν $f(x)$ είναι η αρχική συνάρτηση,

1. Το πρόβλημα ζητά να βρούμε τη σειρά Taylor της συνάρτησης $f(x)=\frac{x}{x^2+4x+5}$ μέχρι και τον τρίτο όρο (τρίτο παράγοντα).\n2. Η σειρά Taylor μιας συνάρτησης $f(x)$ γύρω από

1. The problem asks to identify which function corresponds to the given slope field shown in the top-right graph. 2. A slope field represents the direction of the tangent (slope) t

1. **State the problem:** Find the limit of a function as the variable approaches a certain value. Since the user did not specify the exact function or limit, let's consider a gene

1. **Problem Statement:** We are given the function $$h(x) = x^4 - 3x^3 - 10x + 2$$ and need to find its first and second derivatives, critical points, and plot the function with t

1. Muammo: $y = x^2$, $x = 2$, va $y = 0$ chiziqlari bilan chegaralangan shaklni $Oy$ o‘qi atrofida aylantirishdan hosil bo‘lgan jism hajmini topish. 2. Formulalar va qoidalar: $Oy

1. Muammo: $y = x^2$, $x = 2$, va $y = 0$ chiziqlari bilan chegaralangan shaklni $Oy$ o'qi atrofida aylantirishdan hosil bo'lgan jism hajmini topish. 2. Formulalar va qoidalar: $Oy

1. **Problem Statement:** Find the area of the region bounded by the parabola $y = x^{2}$, the vertical line $x = 2$, and the horizontal line $y = 0$. 2. **Formula and Explanation:

1. **Problem statement:** A window is shaped as a rectangle surmounted by an equilateral triangle. The total perimeter is 12 m. Find the rectangle dimensions that maximize the wind

1. Muammo: Berilgan parametrik tenglamalar $x = 6 \cos^3 t$, $y = 6 \sin^3 t$ va $0 \leq t \leq \frac{\pi}{3}$ oraliqdagi egri chiziqning yoyi uzunligini topish. 2. Egri chiziqning