In this lesson, our instructor Vincent Selhorst-Jones teaches about the Area Under a Curve (Integrals). Youll learn about approximation by rectangles and the various methods to choose them. Youll also go over the rectangle method from the left-most point, right-most point, mid-point, maximum, and minimum. Vincent also teaches how to evaluate the area approximation and how to find area with a limit. Something big will be revealed and then youll practice with four examples.
The integral is a way to find the area underneath some portion of a curve.
We can approximate an integral by using rectangles. We can break an interval up into sub-intervals, and then put a rectangle the width of each sub-interval in place. By taking the area of each rectangle and summing them all up, we have an approximation
of the area under the curve in that interval.
The width of each rectangle is the same, because we cut the sub-intervals evenly. However, the height of each rectangle can vary depending on the height of the function in that sub-interval. Furthermore, the function has different heights throughout
the sub-interval, so we have to come up with some method to choose an xi in each sub-interval to find the height of its associated rectangle: f(xi).
Here are some of the most common methods for choosing the xi in each sub-interval:
Left-Most Point: We choose xi such that it is the left-most point in each sub-interval. The height of the rectangle is based on its left side.
Right-Most Point: We choose xi such that it is the right-most point in each sub-interval. The height of the rectangle is based on its right side.
Mid-Point: We choose xi such that it is the mid-point in each sub-interval. The height of the rectangle is based on its middle.
Maximum (Upper Sum): We choose xi such that the rectangle is the highest possible for the sub-interval. The height of the rectangle is the highest place the function achieves in that sub-interval.
Minimum (Lower Sum): We choose xi such that the rectangle is the lowest possible for the sub-interval. The height of the rectangle is the lowest place the function achieves in that sub-interval.
We can evaluate the area approximation by summing up all of the rectangles. This comes out to be
[However, if you are told to approximate area, you normally won't need the above formula. Just figure out the area for each rectangle, then add them all together.]
Since the above approximation becomes more accurate as n→ ∞, we can take the limit at infinity to find area.
If the above limit exists, it is called the integral from a to b. It is denoted by
The process of finding integrals is called integration.
The really amazing part is that the integral of f(x) is based on the antiderivative of f(x): that is, the derivative process done in reverse on f(x). We can symbolize the antiderivative of f(x) with F(x). With this notation, we have
f(x) dx = F(b) − F(a).
We can see the above is true, because we can think of the area underneath the curve as a function A(x). Notice that the rate of change for the area is based on the height of the function: f(x). Thus, height is the derivative of area, so area is based
on the antiderivative of height.
Area Under a Curve (Integrals)
Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.