1. The problem is to understand the definite integral $\int_a^b f(x)\,dx$, which represents the area under the curve of the function $f(x)$ from $x=a$ to $x=b$. 2. The formula for the definite integral is: $$\int_a^b f(x)\,dx = F(b) - F(a)$$ where $F(x)$ is any antiderivative of $f(x)$, meaning $F'(x) = f(x)$. 3. Important rules: - The integral sums the values of $f(x)$ multiplied by infinitesimally small widths $dx$ between $a$ and $b$. - If $a > b$, then $\int_a^b f(x)\,dx = -\int_b^a f(x)\,dx$. - The integral can be interpreted as the net area, so areas below the x-axis count as negative. 4. To evaluate a definite integral: - Find the antiderivative $F(x)$. - Substitute the upper limit $b$ and lower limit $a$ into $F(x)$. - Subtract $F(a)$ from $F(b)$. 5. Example: If $f(x) = x^2$, then $F(x) = \frac{x^3}{3}$. 6. Then: $$\int_a^b x^2\,dx = \left. \frac{x^3}{3} \right|_a^b = \frac{b^3}{3} - \frac{a^3}{3} = \frac{b^3 - a^3}{3}$$ This process applies to any integrable function $f(x)$.