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 limits \(a\) and \(b\) are the bounds of integration. - The integral calculates the net area, so areas below the x-axis count as negative. 4. To evaluate: - Find the antiderivative \(F(x)\). - Substitute the upper limit \(b\) into \(F(x)\). - Substitute the lower limit \(a\) into \(F(x)\). - Subtract \(F(a)\) from \(F(b)\). 5. This process gives the exact accumulated value of \(f(x)\) over \([a,b]\).