# Ch 10: Area Under the Curve and Integrals

Watch informative video lessons about the area under the curve and integrals. Learn about Riemann sums, integrals, the fundamental theorem of calculus and more.

## Area Under the Curve and Integrals

If you had to write out every term for all of the sums that we deal with in calculus, you'd have time for little else. Luckily, we can use sum notation to write these sums more compactly and save time. We'll teach you how to translate between sum notation and expanded equations, especially when it comes to Riemann sums. These sums are more than just numbers; we can use them to find the areas under all kinds of functions and graphs, even irregular areas that you may see in everyday life.

Taking these sums of areas is all about rectangles and choosing endpoints. If that sounds strange, don't worry; we'll show you how to identify and draw different types of sums, including left, middle and right sums. There's bound to be to be some overestimation and underestimation of areas when we use mere rectangles, but you'll improve upon this method with the trapezoid rule. Our lessons will strive to give you an intuitive understanding of calculus concepts as you see how shapes can fit a curve.

Riemann sums are only the beginning. We want to lead you to the holy grail of calculus: the integral. You'll learn how to wield these powerful tools when you find out how to take the limits of Riemann sums. Definite integrals will be our starting point; you'll see how a Riemann sum approach can be used to calculate them and come away with an understanding of their linear properties.

We can think about functions in parts and we can think about them globally. As you progress through this calculus section, you'll see how the mean value theorem allows you to do both and get a sense of its possible applications. We'll also define the fundamental theorem of calculus, so you won't want to miss that lesson. Finally, we'll examine anti-derivatives to explore the vital relationship among indefinite integrals, definite integrals and derivatives.

Final Exam
Chapter Exam

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