Integration I: An Overview of Integral Calculus

An integral is simply the area under a function $f(x)$. We oftentimes say the area under a curve. We denote the area under the curve $f(x)$ between points $a$ and $b$ as

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Let's say we want to determine the area under $f(x)$ between points $a$ and $b$. We can approximate the area with a bunch of narrow rectangles. Recall that the area of a rectangle is simply the height times the width. If we use some value of $f(x)$ as a height and the width of a rectangle as the width and then sum the areas of the rectangles then we get an integral.

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Exactly how we go from this concept of adding areas of small rectangles to an exact area is the study of the next few lessons. Also, note that this is where our integration symbol $\int$ comes from. The $\sum$ symbol is a summation symbol. If we take the $S$ in the word "Summation" and stretch it out we get $\int$. In other words, our symbol comes from

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There are two types of integrals: definite and indefinite. A definite integral is where we define exactly where on the $x$-axis we want the area. An indefinite integral is where we don't define where on the $x$-axis we want to know the area.

Finally, one of the important theorems you will need to understand is the Fundamental Theorem of Integral Calculus. It is a way of determining the value of a definite integral. It states that

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