Integration I: An Overview of Integral Calculus

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Lesson Summary: 
Integral Calculus is the study of areas under curves and their application to real world problems. The area under a curve between points $a$ and $b$ are denoted by \begin{equation}\mbox{Area between }a\mbox{ and }b=\int_a^bf(x)\,dx.\end{equation} We discuss definite and indefinite integrals and take a quick look at the Fundamental Theorem of Integral Calculus.
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An Overview of Calculus
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Lesson: 

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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Overview of Integral Calculus

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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Integration as sum of rectangles

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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.

Integral Notation Gives Us
Definite Integral

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 An actual number that is the area between $a$ and $b$.
Indefinite Integral

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 A formula we can use later to determine the area between two specific points.

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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where <Sign in to see all the formulas>. The function $F(x)$ is an antiderivative of $f(x)$.