∫ calculus

1. **Problem (a): Find the area enclosed by the curve $y = 4\cos 3x$, the x-axis, and the lines $x=0$ and $x=\frac{\pi}{6}$.** The area under a curve from $x=a$ to $x=b$ is given b

1. **Problem:** Find the total differential $dx$ if $x = y^3 z = \ln(z)$. 2. **Understanding the problem:** The total differential $dx$ of a function $x = f(y,z)$ is given by

1. The problem is to find the indefinite integral of the function $e^x$, which is written as $\int e^x \, dx$. 2. The formula for the integral of the exponential function $e^x$ is:

1. **Problem (a):** Determine $\int (1 - t)^2 \, dt$. 2. **Formula and rules:** Use the power rule for integration: $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$ for $n \neq -1$.

1. **Problem Statement:** (a) Find the indefinite integral $\int (1 - t)^2 \, dt$.

1. Problem (a): Evaluate $\frac{dy}{dx}$ at $x=2.5$ for $y=\frac{2x^2+3}{\ln(2x)}$. Formula: Use the quotient rule for differentiation:

1. **State the problem:** Differentiate the function $y = (2x+3)^8$ with respect to $x$. 2. **Formula used:** We use the chain rule for differentiation, which states:

1. **State the problem:** Differentiate the function $$f(x) = (x+1)^6$$ with respect to $$x$$. 2. **Formula used:** Use the chain rule for differentiation. If $$f(x) = [g(x)]^n$$,

1. **Problem Statement for Q9:** Find the area between the piecewise function $$f(x) = \begin{cases} x^2 & 0 \leq x \leq 1 \\ 2 - x & 1 < x \leq 2 \end{cases}$$ and $$g(x) = x$$ ov

1. **State the problem:** We have the curve given by the function $$y = x^3 - x^2 - 2x$$ and we want to compute two things over the interval $$[-1, 2]$$: a) The definite integral u

1. Problem Q6: Find the area between the curves $T_1(n) = n^2 + 3n$ and $T_2(n) = 2n^2$ from $n=0$ to $n=5$. 2. Formula: The area between two curves $f(n)$ and $g(n)$ over $[a,b]$

1. The problem asks to use the Riemann sum to approximate the integral of a function over an interval. 2. The Riemann sum formula is $$S_n = \sum_{i=1}^n f(x_i^*) \Delta x$$ where

1. **State the problem:** We are given the curve defined by the function $$y = x^3 - x^2 - 2x$$. We need to compute two things over the interval $$[-1, 2]$$: a) The definite integr

1. **Problem (a):** Evaluate $\frac{dy}{dx}$ at $x=2.5$ for $y=\frac{2x^2+3}{\ln(2x)}$. 2. **Formula and rules:** Use the quotient rule for derivatives: if $y=\frac{u}{v}$, then

1. Problem 1: Find the area enclosed between the curves $y = x^2$ and $y = 4x - x^2$. 2. First, find the points of intersection by setting $x^2 = 4x - x^2$.

1. Problem Q6: Find the area between the curves $T_1(n) = n^2 + 3n$ and $T_2(n) = 2n^2$ from $n=0$ to $n=5$. 2. Formula: The area between two curves $y=f(x)$ and $y=g(x)$ over $[a,

1. **Problem Statement:** We want to approximate the area of the region bounded by the graph of $f(x) = \sin(x)$ and the x-axis between $x=0$ and $x=0.5$ using a Riemann sum. 2. **

1. Let's clarify where the constant $a$ comes from when differentiating functions involving $x$. 2. Typically, $a$ is a constant coefficient in a function, for example, $f(x) = a x

1. Let's find the third derivative of the function from the previous question. First, we need to know the original function $f(x)$. Since it was not provided here, I'll assume it w

1. The third derivative of a function is the derivative of the second derivative. It measures the rate of change of the acceleration of the function. 2. To find the third derivativ

1. **State the problem:** Understand what the difference quotient means in mathematics. 2. **Definition:** The difference quotient is a formula that calculates the average rate of