1. The problem asks: Calculate the integral of a function as an example. 2. Let's find the integral of $f(x) = 2x$ over the interval $[0,3]$. 3. The formula for the definite integral is: $$\int_a^b f(x)\,dx$$ which means the area under the curve from $x=a$ to $x=b$. 4. Substitute $f(x) = 2x$, $a=0$, and $b=3$: $$\int_0^3 2x\,dx$$ 5. Find the antiderivative of $2x$, which is $x^2$. 6. Evaluate the antiderivative at the bounds: $$x^2 \Big|_0^3 = 3^2 - 0^2 = 9 - 0 = 9$$ 7. So, the integral equals 9, which represents the area under the curve $2x$ from 0 to 3. 8. This example shows how integrals calculate accumulated quantities like area.