Riemann sums are a simple way to estimate the area under curves that are not in a nice, tidy shape. We can break this area into smaller, easy pieces to work with. Here’s how we can do it step by step:
Divide the Interval: Start by splitting the area between two points, (a) and (b), into (n) equal sections. Each piece will have a width of (\Delta x = \frac{b-a}{n}).
Pick Sample Points: Choose a point in each of these smaller sections. You can pick the left side, the right side, or the middle of each section.
Find the Riemann Sum: Now we can calculate the Riemann sum, which looks like this: [ S_n = \sum_{i=1}^{n} f(x_i^) \Delta x ] Here, (x_i^) is the point you picked in each section.
Take the Limit: If we keep making more and more sections (as (n) gets really big), the Riemann sum gets closer to what's called a definite integral: [ \int_a^b f(x) , dx ]
Using Riemann sums helps us get a good guess of the area under complicated curves. This method is really useful in calculus!
Riemann sums are a simple way to estimate the area under curves that are not in a nice, tidy shape. We can break this area into smaller, easy pieces to work with. Here’s how we can do it step by step:
Divide the Interval: Start by splitting the area between two points, (a) and (b), into (n) equal sections. Each piece will have a width of (\Delta x = \frac{b-a}{n}).
Pick Sample Points: Choose a point in each of these smaller sections. You can pick the left side, the right side, or the middle of each section.
Find the Riemann Sum: Now we can calculate the Riemann sum, which looks like this: [ S_n = \sum_{i=1}^{n} f(x_i^) \Delta x ] Here, (x_i^) is the point you picked in each section.
Take the Limit: If we keep making more and more sections (as (n) gets really big), the Riemann sum gets closer to what's called a definite integral: [ \int_a^b f(x) , dx ]
Using Riemann sums helps us get a good guess of the area under complicated curves. This method is really useful in calculus!