Definite integrals and the Fundamental Theorem of Calculus
Riemann sums, the definite integral as net area, both parts of the Fundamental Theorem of Calculus, area between curves and the average value of a function.
Area by rectangles
How do you find the area under \(y = x^2\) from \(x = 0\) to \(x = 2\)? The top edge is curved, so no formula from geometry applies.
Approximate it with rectangles: split \([0, 2]\) into \(n\) strips, build a rectangle on each one reaching up to the curve, and add their areas. That's a Riemann sum. As \(n\) grows, the rectangles get thinner and the sum approaches the exact area.
The Fundamental Theorem of Calculus
Computing limits of sums would be exhausting. The Fundamental Theorem connects definite integrals to antiderivatives.
Part 1 says integration and differentiation undo each other. Part 2 is the tool for computing.
The \(+C\) isn't needed: it would cancel in \(F(b) - F(a)\).
Net area versus total area
Area between curves
U-substitution with limits
Average value
The average value of \(f\) on \([a, b]\) is \(\frac{1}{b - a}\int_a^b f(x)\,dx\). For example, the average of \(x^2\) on \([0, 2]\) is \(\frac{1}{2} \cdot \frac{8}{3} = \frac{4}{3}\).
Practice
Quick check
If \(F' = f\), what is \(\int_a^b f(x)\,dx\)?
That's Part 2 of the Fundamental Theorem of Calculus.
Quick check
What is \(\frac{d}{dx}\int_0^x \cos(t^2)\,dt\)?
By Part 1, differentiating the integral with upper limit \(x\) gives back the integrand at \(x\).
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