∫ calculus

1. **State the problem:** We need to find the limit of the function $$\frac{x-3}{x-2}$$ as $x$ approaches $2$. 2. **Recall the limit concept:** The limit of a function as $x$ appro

1. **Problem Statement:** (i) Evaluate the double integral $$\iint_D e^{-x-2y} \, dA$$ where $$D$$ is the region in the first quadrant bounded by $$y \leq x$$.

1. **Problem Statement:** Evaluate the line integral $$\int_C x^2y^2\,dx + 4xy^3\,dy$$ where $C$ is the piecewise path consisting of line segments from $(0,0)$ to $(0,3)$, $(0,3)$

1. **Problem (a)(i):** Evaluate $$\iint_D e^{-x-2y} \, dA$$ where $$D$$ is the region in the 1st quadrant with $$y \leq x$$. 2. **Step 1:** Describe the region $$D$$. Since $$D$$ i

1. **State the problem:** Evaluate the line integral $$\int_C x^2 y^2 \, dx + 4xy^3 \, dy$$ where $C$ is the closed path consisting of three line segments: from $(0,0)$ to $(0,3)$,

1. **Problem:** Evaluate the limit $$\lim_{x \to 4} \left( \frac{2x^2 - 4x - 24}{x^2 - 16} - \frac{1}{4 - x} \right)$$ 2. **Formula and rules:** To evaluate limits involving ration

1. Planteamos el problema: calcular la integral $$\int \frac{84x}{4x^4 + 49} \, dx$$. 2. Observamos que el denominador es $$4x^4 + 49$$, que no se factoriza fácilmente con números

1. The problem is to find the integral of the absolute value function $|x|$ with respect to $x$. 2. Recall that the absolute value function $|x|$ is defined as:

1. El problema es evaluar la integral $$\int _{-\infty }^{\infty }e^{-x^{2}}dx$$. 2. Esta integral es conocida como la integral de Gauss y su resultado es $$\sqrt{\pi}$$.

1. **State the problem:** We need to find the limit of the expression $$x - (-1) \cdot \frac{x^3 + 1}{x + 1}$$ as $x$ approaches $-1$. 2. **Rewrite the expression:** The expression

1. **State the problem:** Find the limit $$\lim_{x \to 2} \frac{x^3 - 8}{x - 2}$$. 2. **Recognize the form:** Direct substitution gives $$\frac{2^3 - 8}{2 - 2} = \frac{8 - 8}{0} =

1. **State the problem:** We have the curve given by $$y = 2x^2 + kx - 3$$ and a point on the curve $(3, -6)$. The normal to the curve at this point is parallel to the line $$x + 5

1. **State the problem:** We have a curve defined by $$y = 3x + \frac{12}{x}$$ passing through points A$(2, 12)$ and B$(6, 20)$. We want to find points C and D on the curve where t

1. **Problem statement:** Differentiate each given function. ---

1. **State the problem:** We need to find the integral $$I = \int \tan^{2}(x) \, dx$$. 2. **Recall the identity:** Use the trigonometric identity $$\tan^{2}(x) = \sec^{2}(x) - 1$$.

1. **Tính tích phân** $I = \int \tan^{2}(x) dx$. 2. **Tính tích phân** $I = \int \frac{dx}{x \ln x \ln(\ln x)}$ bằng đổi biến.

1. **State the problem:** We have a rectangle where the length $L$ is 7 times the width $w$, so $L = 7w$.

1. **State the problem:** We have a rectangle where the length $L$ is 7 times the width $W$, so $L = 7W$.

1. **State the problem:** We need to find the area under the curve defined by the function $$f(x) = -0.2x^2 + 2.8x - 1.8$$ from $$x=1$$ to $$x=13$$ using the definite integral $$\i

1. **State the problem:** We need to calculate the double integral $$\iint_R (3xy - x^2 - y^2 + 6) \, dA$$ where the region $R$ is bounded by $x=0$, $y=0$, and $x + 2y = 2$.

1. **Problem Statement:** We are given the graph of a function $f$ and asked to find specific values and characteristics from the graph. 2. **(a) Find $f(-1)$:**