4 min readβ’january 7, 2023

Now that you have mastered derivatives, itβs time to explore another major concept in calculus, the** integral**. In our study of derivatives, we learned that the graph of a derivative shows the rate of change of the function. If we find the area between such a graph and the x-axis, that area gives the *accumulation of change* for the function.Β

To calculate this accumulation of change, weβll use basic geometry area skills: rectangles, circles, triangles, and trapezoids. Letβs say we have a function, g(x), which has a derivative, f(x), as shown below.

To find

we will add up the area between the derivative graph and the x-axis over the interval from 0 to 3 (the lower number on the integral symbol defines the starting point).

The region from x = 0 to x = 1 is a quarter of a circle with a radius of 1. That means it has an area of Ο/4. However, we have to be mindful of the sign of the areas when evaluating integrals this way. As we move along the x-axis to the __right__, the sign of the x-portion of the area is __positive__. If the derivative graph is __below__ the x-axis, the y-portion of the area is __negative.__ In this case, our answer should be -Ο/4 because positive (x-portion) times negative (y-portion) is negative. Similarly, from x = 1 to x = 3 we have a right triangle. Our x-portion is positive and our y-portion is negative. Using the formula for the area of a triangle we have 12(2)(-1)= -1. Adding this to the area from the previous interval, we have

If we move to the *left* along the x-axis, the x-portion of the area becomes *negative*. If the derivative graph is *above* the x-axis, the y-portion of the area is positive. For example, to find

the x-portion of the trapezoid area is negative and the y-portion is positive.

Another way to approximate the area under a curve (i.e., an integral), is to use the area of rectangles or trapezoids using something called a Riemann Sum. Letβs say we have the function, f(x), graphed below.

The true value of

would give us the exact area under the curve (between the function and the x-axis). However, we are unable to use the shapes we used in the previous section to find this exact area. Thatβs where Riemann comes in handy!Β

There are 4 different types of Riemann Sums: right, left, mid-point, and trapezoid. These types of problems specify how many sub-intervals to use (i.e., how many rectangles or trapezoids to create). For this example, letβs assume we are asked to use 5 equal subintervals to estimate the value of

For a right Riemann Sum, we would create five equally-sized bases for rectangles using x-lengths of 1 (e.g. the length from 0 to 1 would be the base of the first rectangle). Then, we would create the height of each rectangle by finding the value of f(x) for the *right* side of each rectangle as shown below. Notice that some of our rectangles will overestimate the area while others will underestimate the area. The **Riemann Sum** comes from adding up the area for all of the rectangles. In this case

For a **Left Riemann Sum**, weβll repeat the process, but use the *left* side of each rectangle to establish the height as shown below. This results in the following estimate:

For a **Midpoint Riemann Sum**, weβll find the height of each rectangle by using the height for the middle of each base as shown below. This results in the following estimation:

The final Riemann Sum is a trapezoidal sum. Recall that the area of a trapezoid is found by

In essence, we are multiplying the base by the average of the two heights. Using the original example function, that gives an estimate of

Note that in the examples above, each rectangle has the same base of length 1. In many problems, the bases will be of different lengths. Donβt forget to change the bases for each rectangle or trapezoid!Β

Finally, for some types of functions, it is possible to determine whether a particular kind of Riemann Sum will lead to an **overestimate** or an **underestimate** of the integral. For *increasing functions,** *a **Right Riemann Sum** will always lead to an overestimate, because the right side of every rectangle will be higher than the actual function for that rectangle. Similarly, for *decreasing functions, *a Right Riemann Sum will always lead to an underestimate because the left side of every rectangle will be lower than the actual function for that rectangle.

These problems take a lot of practice, but once you commit the process for finding delta x and x sub i to memory, they become very doable. Good luck!

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