About Course

Discover how the area under a curve emerges from adding up infinitely many tiny slices, and learn the powerful connection that ties integration to differentiation. In this targeted mini-course, you’ll build a solid understanding of definite integrals through hands-on approximations, then see how the Fundamental Theorem of Calculus bridges two core operations. By the end, you’ll confidently compute exact areas, interpret accumulation in real contexts, and wield the theorem’s two parts as one seamless tool.

What Will You Learn?

  • Build exact area calculations by refining rectangle-sum approximations
  • Distinguish and use left, right, and midpoint summation methods
  • Grasp how the accumulation of change yields a new function
  • Apply antiderivatives to evaluate definite integrals in one step
  • Interpret and solve real-world accumulation and area problems

Course Content

1. Defining the Definite Integral via Riemann Sums
Explore how summing the areas of ever-thinner rectangles beneath a curve reveals the true accumulated area. You’ll start by partitioning intervals, practice approximating with left-, right-, and midpoint rectangles, and then see how pushing the slice width to zero transforms a rough sum into an exact value.

  • 1.1 Partitioning Intervals & Summation Approaches
  • 1.2 Approximating Area with Left, Right & Midpoint Rectangles
  • 1.3 Transitioning from Finite Sums to the Definite Integral Concept

2. The Fundamental Theorem of Calculus: Parts I
Explore how adding up infinitely many tiny slices under a curve builds a new “accumulation” function, and see why the rate at which that accumulation grows always matches the original curve’s height. This first part of the Fundamental Theorem reveals the deep link between area and change, grounding your understanding of why integration and differentiation are inverse processes.

3. Computing Definite Integrals with the Fundamental Theorem, Part II
Learn how knowing any single antiderivative lets you bypass sums entirely and compute exact accumulated area in one swift step. This second part of the Fundamental Theorem turns integration into the simple act of evaluating two values and taking their difference—streamlining everything from physics applications to area calculations.

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