∫ calculus

1. **Problem:** Evaluate the area between the graph of $f(x) = x$ and the x-axis on the interval $[0, 3]$. 2. **Formula:** The area under the curve from $a$ to $b$ is given by the

1. The problem is to find the integral of the function $2x - 2.7x + 7$ with respect to $x$ and verify if it equals $\frac{1}{4}x^4 - 7x + 1$. 2. First, simplify the integrand:

1. **State the problem:** We need to find the indefinite integral of the function $ (3 - x)^2 $ with respect to $ x $. 2. **Recall the formula:** The integral of a function $ f(x)

1. **State the problem:** Evaluate the double integral

1. **State the problem:** We need to find the surface area of revolution of the function

1. The problem is to understand what an ordinal differential equation means and to see a clear example. 2. An ordinary differential equation (ODE) is an equation involving a functi

1. **State the problem:** Find the maximum value of a function (not specified in the question). 2. **General approach:** To find the maximum of a function $f(x)$, we first find its

1. The problem asks to find the limit $$\lim_{x \to 0} \sin\left(\frac{\pi}{x}\right)$$ given that $$f(0.001) = f(0.0001) = 0$$ where $$f(x) = \sin\left(\frac{\pi}{x}\right)$$. 2.

1. **State the problem:** We want to investigate the limit $$\lim_{x \to 0} \sin\left(\frac{\pi}{x}\right).$$ 2. **Recall the function and its behavior:** The function is $$f(x) =

1. The problem is to evaluate the integral $$\int f(\sin \theta \cos \theta) \, d\theta$$. 2. To solve this, we need to understand the function $f$ and the expression inside it, $\

1. **State the problem:** We need to evaluate the integral $$\int \frac{1 + \ln x}{x \ln x} \, dx.$$\n\n2. **Rewrite the integral:** Split the integral into two parts:\n$$\int \fra

1. **Problem statement:** Find the unit tangent vector, unit normal vector, and curvature of the circle defined by $$x = a \cos \theta, y = a \sin \theta, z = 0$$ at the point with

1. **Problem:** Find the natural domain of the function $f(x) = \sqrt{x^2 - 5x + 6}$. 2. **Formula and rules:** The natural domain of a square root function requires the radicand (

1. **State the problem:** Find the limit $$\lim_{x \to -\infty} \sqrt{4x^2 - 6} + \sqrt{x^2 + 1}$$. 2. **Recall the rule:** For limits involving square roots of quadratic expressio

1. **Problem:** Determine the type of discontinuity for the piecewise function $$f(x) = \begin{cases} 2x^3 + x, & x < -2 \\ x^4 - x^2, & x \geq -2 \end{cases}$$ at $x = -2$.

1. **State the problem:** Find the derivative of $A(x)$ at $x=\pi$, where $$A(x) = \int_2^x \frac{\cos t}{1+t} \, dt.$$

1. Differentiate $y = 2x^3 + 5x^2 - 4x + 9$. The derivative of a polynomial is found by applying the power rule: $\frac{d}{dx} x^n = n x^{n-1}$.

1. The problem asks to find the derivative with respect to $x$ of the function $a^3 \log_a x$. 2. Recall the formula for the derivative of $\log_a x$:

1. **State the problem:** We need to evaluate the integral $$\int 15x \sqrt{x} + 4 \, dx$$. 2. **Rewrite the integral:** Recall that $$\sqrt{x} = x^{\frac{1}{2}}$$, so the integral

1. **Problem:** Evaluate the limit $$\lim_{t \to 0} \frac{\sin t}{t}$$ using special limits. 2. **Formula and important rule:** The special limit $$\lim_{x \to 0} \frac{\sin x}{x}

1. The problem is to find the indefinite integral of the function $$f(x) = \frac{2}{\sqrt{x}} + 6x - 3 \cdot 2^{4x} + \left(\frac{1}{3}\right)^{-x}$$ with respect to $x$. 2. Recall