Definite Integral 7209A5

1. The problem is to understand the definite integral $\int_a^b f(x) dx$. 2. The definite integral $\int_a^b f(x) dx$ represents the signed area under the curve $y=f(x)$ from $x=a$ to $x=b$. 3. The fundamental theorem of calculus states that if $F(x)$ is an antiderivative of $f(x)$, then: $$\int_a^b f(x) dx = F(b) - F(a)$$ 4. To evaluate the integral, find an antiderivative $F(x)$ such that $F'(x) = f(x)$. 5. Then substitute the limits $a$ and $b$ into $F(x)$ and subtract: $$F(b) - F(a)$$ 6. This gives the exact value of the definite integral, which is the net area between the curve and the $x$-axis from $a$ to $b$. 7. Important rules: - If $f(x)$ is positive on $[a,b]$, the integral gives the area under the curve. - If $f(x)$ is negative on $[a,b]$, the integral gives a negative value representing area below the $x$-axis. - The integral sums infinitesimal slices $f(x) dx$ over the interval. This is the general explanation of the definite integral $\int_a^b f(x) dx$.