∫ calculus

1. **Problem statement:** Find the point(s) on the curve $x^2 - y^2 = 4$ closest to the point $(6,0)$. 2. **Formula and approach:** The distance squared from a point $(x,y)$ on the

1. مسئله: مشتق تابع را حساب کنید. 2. فرمول مشتق‌گیری: اگر تابعی به صورت $f(x)$ باشد، مشتق آن با نماد $f'(x)$ یا $\frac{df}{dx}$ نشان داده می‌شود.

1. **State the problem:** Find the limit as $x \to 0$ of $$\frac{(\sqrt[4]{1 + x^2} - 1) \sin x}{e^x - 1}.$$\n\n2. **Recall formulas and rules:**\n- For small $x$, $\sin x \approx

1. مسئله را بیان می‌کنیم: باید حد $$\lim_{x \to -1} \frac{2(x+1)^2 - (x+1)^3}{(x+1)^3 - 3(x+1)^2}$$ را بیابیم. 2. ابتدا متغیر کمکی تعریف می‌کنیم: بگذارید $$t = x+1$$. پس حد به صورت

1. **Stating the problem:** Calculate the volume $V$ given by the integral $$V = \frac{\pi}{12} \int_{-2}^{1} \left((9 - x^2)^2 - (x + 7)^2\right) dx$$

1. **Problem 42:** Given the function $$f(x) = \frac{m x^n - 9x^2 + 1}{a x^4 + y x^3 - 2}$$ and the limit $$\lim_{x \to +\infty} f(x) = 3,$$ find the value of $$m \times a$$. 2. **

1. **State the problem:** Convert the double integral $$\int_0^6 \int_{-\sqrt{6x - x^2}}^{\sqrt{6x - x^2}} y \, dy \, dx$$ to polar coordinates. 2. **Understand the region:** The l

1. **State the problem:** We are given the double integral $$\int_{x=-2}^{3} \int_{y=x^2+1}^{x+7} f(x,y) \, dy \, dx$$ and asked to find an equivalent integral with the order of in

1. **State the problem:** A balloon is being filled with helium at a rate of 4 ft³/min. We want to find the rate at which the surface area is increasing when the volume is $$\frac{

1. **Problem statement:** A balloon is being filled with helium at the rate of 4 ft³/min. We need to find the rate at which the surface area is increasing (in ft²/min) when the vol

1. **Problem statement:** A balloon is being filled with helium at the rate of 4 ft³/min. We need to find the rate at which the surface area is increasing (in ft²/min) when the vol

1. **Problem:** Find the exact value of $$\int_1^3 x^2 \ln(3x) \, dx$$ in the form $$a \ln b + c$$ where $$a,c$$ are rational and $$b$$ is an integer. 2. **Formula and rules:** Use

1. **State the problem:** We want to use the Intermediate Value Theorem (IVT) to show that the polynomial $$f(x) = 4x^4 - 9x^2 + 1$$ has a real zero between $$-1$$ and $$0$$. 2. **

1. The problem asks us to use the Intermediate Value Theorem (IVT) to show that the polynomial $$f(x) = 4x^4 - 7x^2 + 2$$ has a real zero between $$-1$$ and $$0$$. 2. The Intermedi

1. **Problem Statement:** Find the approximate area of the shaded region enclosed by the curve $y=3\ln x$ and the line $y=(x-1)\ln 4$ between $x=1$ and $x=4$ using the Trapezoidal

1. نبدأ ببيان المشكلة: نريد إيجاد أفضل تقريب خطي للدالة $f(x) = x - 1 - 2\ln(2)$ عند النقطة $x=1$. 2. صيغة التقريب الخطي (التقريب بالتانجينت) عند نقطة $a$ هي:

1. The problem is to explain why all integrals can be considered as one type of integration. 2. Integration is the process of finding the area under a curve or the accumulation of

1. **State the problem:** We want to express the integrand $$\frac{48x^2}{(x-18)(x+6)^2}$$ as a sum of partial fractions and then evaluate the integral. 2. **Set up the partial fra

1. The problem is to express a given function or expression as an integral. 2. To express a function as an integral, we often use the definition of the integral or known integral f

1. **State the problem:** We want to express the integrand $$\frac{48x^2}{(x-18)(x+6)^2}$$ as a sum of partial fractions and then evaluate the integral. 2. **Set up the partial fra

1. **Problem 1:** Evaluate $$\int \frac{5}{4 - \sqrt{3 - z}} \, dz$$ - Use substitution to simplify the root expression.