∫ calculus

1. **Problem Statement:** Calculate the surface area generated by revolving the curve $f(x) = \sqrt{1-x}$ from $x=0$ to $x=\frac{1}{2}$ around the x-axis. 2. **Formula Used:** The

1. Tentukan turunan pertama \(\frac{dy}{dx}\) dari fungsi dan hitung nilai turunan pada titik yang diberikan. **a.** \(y = 3x^4 - 7x^3 + 4x^2 + 3x - 4\) dengan \(x=2\)

1. Tentukan turunan pertama dari fungsi $y = 3x^{4} - 7x^{3} + 4x^{2} + 3x - 4$ dan hitung nilai turunan pada $x=2$. 2. Rumus turunan fungsi polinomial adalah $\frac{d}{dx} x^n = n

1. **Problem Statement:** Find the total surface area formed by revolving a curve $f(x)$ around the x-axis from point $a$ to $b$.

1. **State the problem:** We have a curve $C$ defined by the implicit equation

1. **Problem Statement:** We are given a piecewise function:

1. **Problem Statement:** Given the piecewise function:

1. **Problem:** Find the derivative of $y = \arcsin(4x^4)$. 2. **Formula:** The derivative of $y = \arcsin(u)$ is $\frac{dy}{dx} = \frac{1}{\sqrt{1-u^2}} \cdot \frac{du}{dx}$.

1. **Problem Statement:** We have a closed cuboid with volume 150 cm³. Height is $h$, width is $w$, and length is $3w$. We want to minimize the surface area.

1. **Problem Statement:** Determine the volume of the solid obtained by rotating the region bounded by the curves $y = x^2 - 2x$ and $y = x$ about the line $y = 4$.

1. We are asked to evaluate the definite integral $$\int_0^2 \sqrt{4 - x^2} \, dx$$. 2. This integral represents the area under the curve of the function $y = \sqrt{4 - x^2}$ from

1. **Problem:** Find the limit $$\lim_{x \to 3} \frac{x^3 - 27}{x - 3}$$. 2. **Formula and rules:** This is a limit of the form $$\frac{f(x) - f(a)}{x - a}$$ where $$f(x) = x^3$$ a

1. **State the problem:** Calculate the surface area formed by revolving the curve $f(x) = \sqrt{1-x}$ from $x=0$ to $x=\frac{1}{2}$ around the x-axis. 2. **Formula used:** The sur

1. **State the problem:** We want to find the integral $$\int \sin^2(3x) \cos^2(2x) \, dx$$. 2. **Recall formulas and identities:** Use the power-reduction formulas:

1. **Problem Statement:** We want to find the surface area of a solid formed by revolving a curve $y=f(x)$ around the x-axis from $x=a$ to $x=b$.

1. **Problem Statement:** We want to find the surface area of a solid formed by revolving a curve $y=f(x)$ around the x-axis from $x=a$ to $x=b$.

1. **State the problem:** We need to determine the domain on which the given function is decreasing based on the graph description. 2. **Understanding decreasing functions:** A fun

1. **Problem:** Find constants $A$ and $B$ such that

1. **State the problem:** Calculate the integral of a function, but since the function is not specified, let's consider a general example: find $\int x^2 \, dx$. 2. **Formula used:

1. We are asked to find the limit: $$\lim_{x \to 3} \frac{2x^2 - 5x - 3}{x-3}$$ 2. The formula for limits involving rational functions is to first try direct substitution. If direc

1. **Problem:** Find the derivative of $f(x) = 2x^3 - 5x^2$. 2. **Formula:** The derivative of $x^n$ is given by $\frac{d}{dx} x^n = nx^{n-1}$.