Integrals

About this course

The second big idea of calculus is area under a curve, built from thin rectangles. The course starts with y = x² on [0, 2] and four left rectangles. Because x² is increasing there, each left rectangle sits under the curve and each right rectangle pokes above it, so L₄ underestimates, R₄ overestimates, and the true area is between 1.75 and 3.75. In the lab you double n a few times and watch both sums close in on 8/3 ≈ 2.667.

The five units are approximating area with rectangles, Riemann sums and the midpoint rule, the definite integral, the Fundamental Theorem of Calculus, and net change and average value. Sigma notation writes the long sums compactly. The midpoint unit shows that each time n doubles the endpoint errors roughly halve, while the midpoint error drops to about a quarter: 0.042, 0.010, 0.0026, 0.00065. The definite integral is the limit of such sums, and it measures net signed area: for 4 − x² on [0, 3], the 16/3 above the axis minus the 7/3 below gives 3. The Fundamental Theorem then connects what looked like separate subjects, slopes and areas, as inverse processes. One arch of y = sin x, from 0 to π, has area exactly 2, and eight midpoint rectangles give about 2.013.

The last unit reads the theorem as net change. For v(t) = t² − 4t + 3 on [0, 4], a made-up cart, three pieces of area 4/3 give a displacement of 4/3 but a distance of 4. A dropped rock has v(t) = −32t, so the displacement from t = 0 to 2 is found by integrating the rate. The course also tells you to check an antiderivative by differentiating it: (2/3)x^(3/2) differentiates to √x.