# What is Integration using rectangles?

Usually, integration using rectangles is the first step for learning integration. At its most basic, integration is finding the area between the x axis and the line of a function on a graph - if this area is not "nice" and doesn't look like a basic shape (triangle, rectangle, etc.) that we can easily calculate the area of, a good way to approximate it is by using rectangles.

Let's take an example:

Imagine, now, that we didn't have a formula for calculating the area of a triangle, but we did have a formula for calculating the area of a rectangle (which is base x height, as usual). Imagine that we want to "guess" the area of that triangle. So we draw ten rectangles, each with base 0.7. The height is whatever the value of x is at the left-hand side of that rectangle. So:

Eventually, we'll get ten rectangles. We can easily calculate the areas of these rectangles:

The areas of the ten rectangles, in order, are:

Note: you don't always have to choose the left-hand side. You can choose the right-hand side, or the midpoint, but make sure you do the same thing for the entire question - don't take one rectangle from the left-hand side and one from the right.

Also note: the smaller the rectangles, the more precise your answer!

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Integration using rectangles, also known as Riemann sums, is a method of approximating the area under a curve by dividing it into rectangles and summing up the areas of these rectangles. This method is foundational to the concept of definite integration.

There are three common types of Riemann sums:

- Left Riemann Sum: The height of each rectangle is determined by the function value at the left endpoint of each subinterval.
- Right Riemann Sum: The height of each rectangle is determined by the function value at the right endpoint of each subinterval.
- Midpoint Riemann Sum: The height of each rectangle is determined by the function value at the midpoint of each subinterval.

As the number of rectangles increases, the approximation becomes more accurate, approaching the actual area under the curve. This process is formalized in the definition of a definite integral as the limit of Riemann sums as the width of the rectangles approaches zero.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- How do you find the area using the trapezoidal approximation method, given #f(x)=5 sqrt(1+sqrt(x)) #, on the interval [0, 4] with n=8?
- How do you use the midpoint rule to approximate the integral #-3x-8x^2dx# from the interval [-1,4] with #n=3#?
- How would I write the Riemann sum needed to find the area under the curve given by the function #f(x)=5x^2+3x+2# over the interval [2,6]?
- How do I find the riemann sum of #y = x^2 + 1# for [0,1] at infinitely small intervals?
- How do you use the Trapezoidal Rule with n=4 to approximate from [2,3] of # 1/(x-1)^2 dx#?

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