∫ calculus

1. **State the problem:** We need to find the limit $$\lim_{x \to -\infty} f(x)$$ where $$f(x) = \begin{cases} 2x^2 + 5, & x < 0 \\ \frac{3 - 5x^3}{1 + 4x + x^3}, & x \geq 0 \end{c

1. **State the problem:** We need to find the limit $$\lim_{x \to -1^+} g(x)$$ where $$g(x) = \frac{4x + 3}{x^2 - 2x - 3}$$. 2. **Factor the denominator:** The denominator is a qua

1. Verify Rolle's theorem for $f(x) = x^2 - 3x + 4$ on $[1,2]$. Step 1: Check if $f(1) = f(2)$.

1. **State the problem:** Find the area of the region bounded by the curves $$y = x^2 - 2$$ and $$y = x$$ between the points $(-1, -1)$ and $(2, 2)$. 2. **Find the points of inters

1. **State the problem:** Evaluate the definite integral $$\int_1^5 \frac{x}{\sqrt{2x - 1}} \, dx.$$\n\n2. **Substitution:** Let $$u = 2x - 1,$$ so that $$du = 2 \, dx$$ or $$dx =

1. **State the problem:** Evaluate the definite integral $$\int_0^1 x(x^2 + 1)^3 \, dx.$$\n\n2. **Use substitution:** Let $$u = x^2 + 1$$ so that $$du = 2x \, dx$$ or $$x \, dx = \

1. **State the problem:** We are given the velocity function of a particle as $v(t) = t^3 - 10t^2 + 29t - 20$ feet per second, and we need to find the displacement of the particle

1. **State the problem:** We need to find the total amount of water that flows out of the faucet during the first two minutes. The flow rate is given by the function $$v(t) = t^3 -

1. The problem asks us to evaluate the definite integral $$\int_0^5 |2x - 5| \, dx$$. 2. To handle the absolute value, find where the expression inside changes sign: solve $$2x - 5

1. **State the problem:** Find the exact value of the definite integral $$\int_1^4 5x \, dx$$. 2. **Set up the integral:** The integral of a function $$f(x) = 5x$$ from 1 to 4 is g

1. The problem asks us to evaluate the definite integral $$\int_0^{\frac{\pi}{2}} 3 \sin x \, dx$$. 2. We start by factoring out the constant 3 from the integral:

1. **Problem 1: Find the area enclosed by the curves** $y = x^2$ and $y^2 = 8x$. 2. First, rewrite $y^2 = 8x$ as $y = \pm \sqrt{8x} = \pm 2\sqrt{2x}$.

1. We are asked to evaluate the definite integral $$\int_0^1 x^2 e^{2x} \, dx$$. 2. To solve this integral, we use integration by parts. Let:

1. **State the problem:** We need to solve the differential equation $$\frac{dy}{dx} = x^2 - 2$$ with the initial condition $$y(3) = 7$$. 2. **Integrate the differential equation:*

1. The problem is to simplify the differential equation $$\frac{dy}{dx} = \frac{1}{3y}$$ and find the implicit solution. 2. Start by separating variables: multiply both sides by $$

1. **State the problem:** Evaluate the triple integral $$\int_0^1 \int_0^1 \int_0^{x+y} \ln z \, dz \, dy \, dx.$$\n\n2. **Integrate with respect to $z$: ** We first compute $$\int

1. Problem statement: A swimming pool is 20 ft wide and 40 ft long and its cross-section has shallow depth 3 ft and deepest depth 9 ft, with the cross-section vertices at $ (0,9),

1. **State the problem:** We have a swimming pool with varying depth and a trapezoidal cross-section. The pool is being filled at a rate of 0.8 ft³/min. We want to find how fast th

1. **State the problem:** We have a cost function $C(x) = x^3 - 3x^2 + 4x$ where $x$ is in hundreds of micro-components and $C(x)$ is in hundreds of dollars.

1. **State the problem:** We need to find the intervals where the function $$k(x) = \frac{1}{x^{2}+3}$$ is convex downward. 2. **Recall the definition:** A function is convex downw

1. **State the problem:** We need to find the interval where the function $$f(x) = x^3 - 6x^2 + 9x + 1$$ is convex upward (concave up). 2. **Recall the definition:** A function is