Questions: Math 172: Integration July 30, 2020 1 "Area under the curve" Integration is often introduced as a way to solve the problem of calculating areas of objects with curved boundaries. That problem is then set up as a problem involving the graph of a function. So integration becomes an operation on functions. It turns out to have many more uses than expressing "area under the curve". Here is a sketch of how integration is typically defined along these lines. Suppose f is a real-valued function with domain containing a closed interval [a, b]. We will determine a way to find the area between the graph of f and the x-axis, from x=a to x=b : We'll start by approximating this area as a sum of areas of rectangles (somewhat like the picture suggests). Let's start with an example. Say f(x)=x^2. Say we want to find the area of the striped region (from x=1 to x=3, above the x-axis, and below the graph):

Transcript text: Math 172: Integration July 30, 2020 1 "Area under the curve" Integration is often introduced as a way to solve the problem of calculating areas of objects with curved boundaries. That problem is then set up as a problem involving the graph of a function. So integration becomes an operation on functions. It turns out to have many more uses than expressing "area under the curve". Here is a sketch of how integration is typically defined along these lines. Suppose $f$ is a real-valued function with domain containing a closed interval $[a, b]$. We will determine a way to find the area between the graph of $f$ and the $x$-axis, from $x=a$ to $x=b$ : We'll start by approximating this area as a sum of areas of rectangles (somewhat like the picture suggests). Let's start with an example. Say $f(x)=x^{2}$. Say we want to find the area of the striped region (from $x=1$ to $x=3$, above the $x$-axis, and below the graph): 1