∫ calculus

1. The problem asks for the area of the region bounded by the curve $y=-x^2 - x - 2$, the x-axis ($y=0$), and the vertical lines $x=-2$ and $x=2$. 2. First, find where the parabola

1. **Given Problem:** Find the derivative of the vector function $$F(t) = \sin(t)\mathbf{i} + t^4\mathbf{j} - e^{2}\mathbf{k}$$

1. **State the problem:** We need to find the arc length of the curve given by the parametric equations $$x = e^t \sin(t), y = e^t \cos(t), z = 9$$ between $$t=0$$ and $$t=4$$. 2.

1. Problem 2.1: Find the constant $C$ such that $$\int_1^4 k(x)\,dx = f(4) + C$$ given that $k(x) = \frac{df}{dx}$. Step 1. Recognize that $k(x)$ is the derivative of $f(x)$, i.e.

1. The statement $\lim_{x \to b} f(x) = K$ means that as the variable $x$ approaches the value $b$, the function $f(x)$ gets arbitrarily close to the number $K$. This describes the

1. The statement $\lim_{x \to b} f(x) = K$ means that as the variable $x$ gets arbitrarily close to the value $b$, the function $f(x)$ approaches the value $K$. This means $f(x)$ c

1. **State the problem:** We want to find the arc length $L$ of the curve $$\mathbf{C}: x=\sin(3t)-3t\cos(3t),\quad y=3t\sin(3t)+\cos(3t),\quad z=4t^2$$

1. **State the problem:** We are given the function $$f(x) = 5x^{\frac{3}{2}} - 3x^{\frac{5}{2}}$$ and need to find the intervals where it is increasing and where it is decreasing.

1. **Problem Statement:** Find the derivative of the vector function $$F(t)=e^{9t} \mathbf{i} + \sin^8(t) \mathbf{j} - \cos(4t) \mathbf{k}$$

1. The problem has three parts. (a) Differentiate $2^{\cos^2 x}$ with respect to $\cos^2 x$.

1. The problem asks for the area of the shaded region bounded by the curves $y^2 = x$, the vertical line $x=4$, and the $x$-axis (which is $y=0$). 2. We first express the region in

1. **State the problem:** Differentiate the function $f(x) = \sin(x^2 + 5)$. 2. **Recall the differentiation rule:** The derivative of $\sin(u)$ with respect to $x$ is $\cos(u) \cd

1. The problem is to differentiate the function $f(x) = x^2 \sin(x^2 + 5)$.\n\n2. We will use the product rule for differentiation, which states that if $f(x) = u(x)v(x)$, then $f'

1. Stating the problem: Differentiate the function $f(x) = x^2 \sin(x^2)$.\n\n2. Use the product rule: If $f(x) = u(x)v(x)$, then $f'(x) = u'(x)v(x) + u(x)v'(x)$. Here, $u(x) = x^2

1. Stating the problem: Differentiate the function $y=(2x-1)(4x+3)$ with respect to $x$. 2. Use the product rule for differentiation: If $y = u \, v$, then $\frac{dy}{dx} = u'v + u

1. The problem is to differentiate the function $f(x) = 5x^6$ with respect to $x$. 2. Apply the power rule of differentiation, which states that if $f(x) = ax^n$, then $f'(x) = a n

1. The problem asks if the function defined on the interval $(1,2]$ has a maximum value. 2. To determine this, we need to know the function's behavior on the interval $(1,2]$.

1. **State the problem:** Prove that the curve defined parametrically by $$x=a[\cos(\theta) + \theta \sin(\theta)], \quad y=a[\sin(\theta) - \theta \cos(\theta)]\n$$ for $a > 0$ an

1. We are asked to find the antiderivative of \(4x \cos(2 - 3x)\) using integration by parts if appropriate.\n\n2. Let \(u = 4x\) and \(dv = \cos(2 - 3x) dx\). Then \(du = 4 dx\) a

1. The problem is to evaluate the double integral $$\int_3^5 \int_1^2 x^2 y \, dy \, dx$$. 2. First, integrate the inner integral with respect to $y$, keeping $x$ fixed:

1. The problem is to find the derivative of the function $$f(x)=\frac{1-\cos(0x)}{1-(1+\tanh^5(x))}$$ with respect to $x$. 2. Simplify the function first: