∫ calculus

1. **State the problem:** We need to find the area of the region bounded by the curve $y = x^2 - 1$ and the x-axis. 2. **Identify the points of intersection:** The region is bounde

1. **State the problem:** We need to find the area of the region bounded by the curve $y = x^2 - 1$ and the x-axis. 2. **Identify the points of intersection:** The region is bounde

1. **State the problem:** We need to find the area of the region bounded by the curve $y = x^3 - 4x$ and the x-axis. 2. **Find the points where the curve intersects the x-axis:** S

1. **State the problem:** We need to find the area of the region bounded by the curve $y = x^2 - 4$, the x-axis ($y=0$), and the vertical lines $x=0$ and $x=2$. 2. **Understand the

1. **State the problem:** Find the area of the region bounded by the curve $y=4-x^2$, the x-axis, and the vertical lines $x=0$ and $x=2$. 2. **Formula and rules:** The area under a

1. **State the problem:** We need to find the area of the region bounded by the curve $y = x^2$, the x-axis, and the vertical lines $x=1$ and $x=3$. 2. **Formula used:** The area u

1. The problem asks for the integral of $\sin(3x + 2)$ with respect to $x$. 2. Recall the formula for integrating sine of a linear function: $$\int \sin(ax + b) \, dx = -\frac{1}{a

1. **State the problem:** We are given the function $$f(x) = (2x - 5)e^x$$ and need to find the exact coordinates of the minimum turning point A. 2. **Find the derivative:** To fin

1. Diberikan integral $$\int \cos x \cdot \sin^3 x \, dx$$. Kita diminta mencari hasil integral tersebut. 2. Gunakan substitusi: misalkan $$u = \sin x$$, maka $$du = \cos x \, dx$$

1. **State the problem:** Find the constant $A$ in the derivative $$\frac{dy}{dx} = \frac{Ax^2 + 12}{x^4 (x^2 - 4)^{1/2}}$$ for the function $$y = \frac{(x^2 - 4)^{1/2}}{x^3}$$ whe

1. Diberikan integral \( \int_1^3 (2x + 3) \, dx \). Kita diminta mencari nilai integral tersebut. 2. Rumus integral dasar yang digunakan adalah \( \int (ax + b) \, dx = \frac{a}{2

1. The problem asks to find the value of the definite integral $$\int_0^2 (x^2 - 2x + 1) \, dx$$. 2. The formula for the definite integral of a function $f(x)$ from $a$ to $b$ is:

1. Diberikan integral $$\int_0^{\frac{\pi}{2}} \sin x \, dx$$. Kita diminta mencari nilai integral tersebut. 2. Rumus dasar integral fungsi sinus adalah $$\int \sin x \, dx = -\cos

1. Diberikan integral $$\int_0^{\frac{\pi}{3}} \cos x \, dx$$. Kita diminta mencari nilai integral tersebut. 2. Rumus dasar integral fungsi kosinus adalah $$\int \cos x \, dx = \si

1. The problem states that the gradient (derivative) of the curve at any point $(x,y)$ is given by $2x - 4$. 2. We need to find the equation of the curve $y=f(x)$ such that its der

1. We are asked to find the area between the curve and the x-axis on the interval $[0,b]$ using definite integrals. 2. The formula for the area under a curve $y=f(x)$ from $x=a$ to

1. **State the problem:** Find the area under the curve $y=3x^2$ from $x=0$ to $x=b$ using a definite integral. 2. **Formula:** The area $A$ under a curve $y=f(x)$ from $x=a$ to $x

1. **State the problem:** We analyze the curve given by the function $$y = \frac{x^2 - 49}{x^2 + 5x - 14}$$ to find its domain, derivatives, intervals of increase/decrease, concavi

1. **State the problem:** We want to simplify the function $$y = \frac{x^2 - 49}{x^2 + 5x - 14}$$ to differentiate it, considering the domain restrictions where the denominator is

1. **State the problem:** We need to find the partial derivative $\frac{\partial z}{\partial y}$ from the implicit equation $$x^2 - 3yz^2 + xyz - 2 = 0.$$\n\n2. **Recall the formul

1. **State the problem:** We are given the implicit function $$x^2 - 3yx^2 + xyz - 2 = 0$$ and asked to find the partial derivative $$\frac{\partial z}{\partial y}$$. 2. **Recall t