∫ calculus

1. **Problem Statement:** Evaluate the double integral $$\iint_R xy \, dA$$ where the region $$R$$ is bounded by the x-axis, the line $$x=0$$, and the curve $$x^2=4a$$. 2. **Unders

1. **Problem statement:** We want to find the relative dimensions (radius $r$ and height $h$) of a right circular cylinder with a closed top that maximize the volume, given a fixed

1. **Problem statement:** We want to find the relative dimensions (radius $r$ and height $h$) of a right circular cylinder with a closed top that maximize the volume, given a fixed

1. The problem is to find the limit: $$\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^3 + 3$$. 2. Recall the limit rule: as $n$ approaches infinity, $\frac{1}{n}$ approaches 0.

1. **Problem statement:** Given the function $$f(x) = x e^{-\frac{x^2}{2}}$$ for $$-2 \leq x \leq 2$$, find the global maximum and minimum values. 2. **Formula and rules:** To find

1. **Problem Statement:** Determine on which intervals the function $f(x) = |9 - x^2|$ is decreasing. 2. **Understanding the function:** The function is the absolute value of $9 -

1. We are asked to find the absolute minimum value of the function $$f(x) = x^3 - 3x$$ on the interval $$[0,2]$$. 2. To find absolute extrema on a closed interval, we check the cri

1. The problem asks why the partial derivative of $y$ with respect to $x$ for the function $-6y - 2x$ equals $-6$. 2. The function given is $f(x,y) = -6y - 2x$.

1. **Problem statement:** Find the absolute minimum value of the function $$f(x) = x \sin x + \cos x$$ on the interval $$[0, \pi]$$. 2. **Formula and rules:** To find absolute extr

1. **Problem Statement:** Find the absolute maximum value of the function $$f(x) = \frac{\ln(x)}{x}$$ on the interval $$[1, e^2]$$. 2. **Formula and Rules:** To find the absolute m

1. **State the problem:** Find the absolute maximum value of the function $f(x) = xe^{-x}$ on the interval $[0,2]$. 2. **Formula and rules:** To find absolute extrema on a closed i

1. **State the problem:** We need to check if the function $$f(x) = \begin{cases} x^2, & x \neq 2 \\ 5, & x = 2 \end{cases}$$ is continuous at $$x=2$$. 2. **Recall the definition o

1. **State the problem:** We want to find for which values of $k$ the function $f(x) = x^3 - 6x^2 + kx + 2$ is one-to-one. 2. **Recall the rule for one-to-one functions:** A functi

1. **State the problem:** We have the function $$f(x) = 1 - 2kx + kx^2 - x^3$$ and we want to find the range of values for the parameter $$k$$ such that $$f(x)$$ is decreasing on i

1. The problem is to test the continuity of the function $f(x) = \ln(x)$ at $x=0$. 2. Recall that the natural logarithm function $\ln(x)$ is defined only for $x > 0$.

1. **Problem Statement:** Find the two critical values of the function $f(x) = 2x^3 - 18x^2 + 30x - 11$. 2. **Recall:** Critical values occur where the derivative $f'(x)$ is zero o

1. **Problem Statement:** Find the critical values, intervals of increase and decrease, and relative maxima and minima for the cost function $$C(x) = -0.1x^2 + 18x - 340$$ on the d

1. The problem asks to list the x-values of all relative maxima and minima of the function $$C(x) = -0.1x^2 + 18x - 340$$ on the interval $$0 < x < 400$$. 2. To find relative maxim

1. **State the problem:** We want to find the limit $$\lim_{x \to 0^+} \left(e^x + x\right) \frac{1}{x}$$ as $x$ approaches $0$ from the right. 2. **Recall the limit properties and

1. **State the problem:** Find the limit $$\lim_{x \to 0^+} \left(e^2 + x\right) \frac{1}{x}$$ as $x$ approaches $0$ from the right. 2. **Recall the limit properties:** When evalua

1. **State the problem:** Find the limit $$\lim_{x \to 0^+} \frac{e^x + x - 1}{x}$$. 2. **Recall the formula and rules:** The limit involves an indeterminate form of type $$\frac{0