1. The problem: Understand what an integral is and how it is used in calculus. 2. Definition: An integral is a mathematical tool used to find the area under a curve or to accumulate quantities. 3. There are two main types of integrals: definite and indefinite. 4. The indefinite integral (antiderivative) of a function $f(x)$ is written as $$\int f(x)\,dx$$ and represents a family of functions whose derivative is $f(x)$. 5. The definite integral from $a$ to $b$ is written as $$\int_a^b f(x)\,dx$$ and gives the net area between the curve $y=f(x)$ and the $x$-axis from $x=a$ to $x=b$. 6. Important rules: - Linearity: $$\int (af(x) + bg(x))\,dx = a\int f(x)\,dx + b\int g(x)\,dx$$ - Power rule for integration: $$\int x^n\,dx = \frac{x^{n+1}}{n+1} + C$$ for $n \neq -1$ 7. Example: Find the indefinite integral of $f(x) = 3x^2$. 8. Using the power rule: $$\int 3x^2\,dx = 3 \int x^2\,dx = 3 \cdot \frac{x^{3}}{3} + C = x^3 + C$$ 9. Explanation: We increased the power by 1 and divided by the new power, then multiplied by the constant. 10. For definite integrals, evaluate the antiderivative at the upper and lower limits and subtract: $$\int_a^b f(x)\,dx = F(b) - F(a)$$ where $F(x)$ is the antiderivative of $f(x)$. This is the fundamental concept of integrals in calculus.