∫ calculus

1. **State the problem:** Find the volume of the sphere defined by the equation $$x^2 + y^2 + z^2 = a^2$$ using triple integration. 2. **Set up the integral:** The sphere is symmet

1. The problem appears to involve the expression \(\sin 7 \cos 3 x x dx\), which is ambiguous. Assuming you want to integrate the function \(f(x) = \sin(7) \cos(3x) x\) with respec

1. The problem is to evaluate the integral $$\int \frac{2x+1}{4x^2+13} \, dx$$. 2. Notice that the denominator is a quadratic expression $4x^2 + 13$ which cannot be factored easily

1. **State the problem:** We want to evaluate the integral $$\int \sqrt{x^3 - 3} (x^2 - 1) \, dx.$$\n\n2. **Substitution:** Let $$u = x^3 - 3.$$ Then, differentiate both sides with

1. **State the problem:** We need to evaluate the integral $$\int \sqrt{x^3 - 3} (x^2 - 1) \, dx.$$\n\n2. **Rewrite the integral:** Let $$I = \int \sqrt{x^3 - 3} (x^2 - 1) \, dx.$$

1. **State the problem:** We are given the function $f(x) = x^3 + 3x^2 - 4x$ with roots at $x = -4, 0, 1$. We need to find the maximum and minimum points of this function and sketc

1. Enunciado del problema: Calcular la integral $$\int t \arcsin(t) \, dt$$. 2. Para resolver esta integral, usaremos integración por partes. Recordemos la fórmula:

1. **State the problem:** We are given the function $$y=\frac{1}{x}$$ and asked to find the open intervals where the function is increasing, decreasing, or constant. 2. **Analyze t

1. **State the problem:** Given parametric equations $x = k(t - \sin t)$ and $y = k(1 - \cos t)$ with $k \neq 0$, find the second derivative $\frac{d^2 y}{dx^2}$ at $t = \frac{\pi}

1. **State the problem:** Evaluate the limit $$\lim_{x \to \infty} \frac{8x^3 - 7x^2 + 2x + 1}{2x^3 + 5x^2 - x - 1}$$. 2. **Identify the highest power of $x$ in numerator and denom

1. The problem asks to find $\frac{dx}{dt}$ when $x = 7 \ln t$. 2. Recall that the derivative of $\ln t$ with respect to $t$ is $\frac{1}{t}$.

1. The problem is to find the derivative of a function, but the function is not specified. 2. To find the derivative, we need the function expression, for example, if the function

1. **Problem a)**: Evaluate or describe the region of integration for $$\int_0^4 \int_{\frac{4-y}{7}}^1 f(x,y) \, dx \, dy.$$

1. **State the problem:** Evaluate the definite integral $$\int_0^\pi \sin^2(t) \cos(t) \, dt$$. 2. **Use substitution:** Let $$u = \sin(t)$$, then $$du = \cos(t) dt$$.

1. Given integrals: \(\int_0^4 x^3 dx = 60\), \(\int_2^4 x dx = 6\). 2. (a) \(\int_0^2 x^2 dx = \left[ \frac{x^3}{3} \right]_0^2 = \frac{8}{3} \approx 2.6667\).

1. The problem is to find the indefinite integral of the polynomial function $$3x^2 + 7x - 2$$ with respect to $$x$$. 2. Recall the power rule for integration: $$\int x^n \, dx = \

1. The problem is to solve the differential equation $$\frac{dy}{dx} = 5y^2 \cos(x)$$. 2. This is a separable differential equation. We can rewrite it as:

1. The problem asks to find the value of $f(0)$ given the derivative $f'(x) = -10x^4 + 8x^3$ and the value $f(1) = 9$. 2. To find $f(0)$, we first need to find the original functio

1. The problem is to solve the differential equation $$\frac{dy}{dx} = 3x^{-2} e^{-y}$$ for $y$ as a function of $x$. 2. Separate variables to isolate $y$ terms on one side and $x$

1. Determine o domínio da função $$z = \frac{1}{\sqrt{1 - x^2 - y^2}}$$. - Para que a função esteja definida, o denominador não pode ser zero e o radicando deve ser positivo.

1. Problem: Find $$\lim_{x \to 16} \frac{x - \sqrt{x-12}}{x - 16}$$. Step 1: Direct substitution gives $$\frac{16 - \sqrt{16-12}}{16 - 16} = \frac{16 - 2}{0} = \frac{14}{0}$$ which