Riemann Sums and FTC

This unit covers finding area under the curve using Riemann Sums, The Fundamental Theorem of Calculus, Properties of Integration, U-Substitution and Indefinite Integrals.

For Calculus, when estimating area under curves, you will be asked to use Riemann Sums. Riemann Sums will be estimated using 4 different options:

1. Left Riemann Sum
2. Right Riemann Sum
3. Midpoint Riemann Sum
4. Trapezoidal Riemann Sum

When dealing with Riemann Sums, you will be asked to divide up your rectangles or trapezoids. By dividing these up, this is called partitioning. When you partition your Riemann Sums, you are finding the width of the rectangle or trapezoid. This is called delta x. Delta is represented by a triangle. (🔺x)

Suppose f is a function given on an interval (a, b). To find delta x, you subtract b from a and divide by the n value, which is the number of rectangles or trapezoids the problem is asking for.

When doing a left, right, or midpoint Riemann Sum, you will be using rectangles. The width of the rectangle is the delta x. The height of the rectangle will come from the y-value of the rectangle either at the left point, the right point, or the midpoint of the rectangle.

The Trapezoidal Riemann Sum will give the closest estimate to the actual area under the curve. When you draw the trapezoids, you will find the height of each x-value. With the exception of the first y-value and the last y-value, you will double each of the y-values and then add them all together. Then you will multiply that answer by (b-a) divided by (2n), where the n is the number of trapezoids that you used!

If a function is increasing, then:

• A left Riemann Sum under estimates the area
• A right Riemann Sum over estimates the area

If a function is decreasing, then:

• A left Riemann Sum over estimates the area
• A right Riemann Sum under estimates the area

If a function is concave up, then a trapezoidal sum will over estimate the area.

If a function is concave down, then a trapezoidal sum will underestimate the area.

If a function is continuous on a closed interval, then the definite integral exists.

If a function is bounded on a closed interval and as at most a finite number of points of discontinuity then the definite integral on that closed interval will exist.

The Fundamental Theorem of Calculus (FTC), is aptly named because it links the two branches of calculus together: Differential Calculus and Integral Calculus.

FTC actually has two different parts, each of which relates an integral to an antiderivative.

In order to be able use part 1 of FTC, you need to have a function that is continuous on a closed interval [a, b]. If you change the b to x, then you will run an integral, F(x), from [a, x] instead of [a, b].

Part 1 of FTC states that if we differentiate F(x) with respect to x, we will get the original function back. That means that the integral from [a, x] of f(t) dt is an antiderivative of f.

If your interval is from [a, x], then you will not have to use the chain rule when you use FTC part 1. But, there are some rules you will need to follow if your integral is not running from [a, x].

Three different rules:

Process when [a, b] is actually [g(x), h(x)]:

Part 1 of the Fundamental Theorem of Calculus establishes a relationship between the derivative and the definite integral. Part 2 of FTC provides a method for finding a definite integral without using Riemann Sums.

When you integrate over an interval, [a, b], you will always get a numerical answer.

When an integral does not have any boundaries, it is called the indefinite integral. Everything will work exactly like it did with the definite integrals, but your answer will not be a numerical value. It will be the original equation + C. The + C is called the constant of integration.

When you integrate, you want to be able to use the table of integration as much as possible. But sometimes, that is not an option. When you are not able to do an integration using one of the forms from that table of integrals, you will want to use u-substitution to try and transform the integral into something that you are able to use.

The method of u-substitution is based on the Chain Rule.

Given: