Calculus Explained in 5 Minutes Continued: Integration & Kinematics

Written By: Mika Lai

• If you have not read the previous article which introduces you to the basics of calculus, it is best that you read the first article before delving into the concepts of integration.

Integration: The Antiderivative

Although you have probably heard of differentiation, the term integration may be even more unfamiliar. But to state it simply, integration means the antiderivative- the opposite- it is when you reverse the steps of differentiation to find the area below the curve at 2 given points.

Although this example is displayed in an unknown format, think of each unknown symbol like a new letter to the alphabet that will help you spell a new word.

Although this example is displayed in an unknown format, think of each unknown symbol like a new letter to the alphabet that will help you spell a new word.

What this means is that it is asking for the area under the curve between x = 2 and x = 1.

First we need to integrate the equation.

This notation is called the indefinite integral, as there are no values you need to plug in. As you might have noticed, we apply the chain rule to differentiate the equation, so now we have to reverse the steps.

Instead of subtracting the exponent value by 1, we add 1, then we take this new exponential value and instead of multiplying, we divide. In this example the integral of the equation would be:

One very important step is to add a constant. This is because when you derive a constant the value is always 0, so there is no way of knowing what that constant could be. However if you had a coordinate of the graph, you can always plug it in to solve for C.

Now you still need to find the area, as this is just the integral.

Now all you need to do is plug in the two values and subtract outcomes. Always start with the value at the top(the larger value). Because of this method, the constant makes no difference, as it will be subtracted and will equal 0, and so in questions similar to this example, you don’t need to bother solving for the constant to be able to find the area.

• Note that when working with trigonometry in calculus, your units should always be in radians and not degrees, so don’t forget to switch your calculator to the radians setting.

Here is a more complicated example:

One final example, and you will have mastered the basics of calculus.

Try to figure out the answer yourself first before you look at the working and final answer underneath.

1. Separate & integrate each function separately

2. If the derivative of sin(x) = cos(x), the integral of cos(x) would be sin(x)

3. Remember, in logarithms, the derivative is the same, so the integral would be the same as well, but since the exponential is 4x, we still have to apply the chain rule.

4. When done the integral of this would be:

5. Again if we integrate this separately, the integrals should be:

6. The final answer would be:

We can check if the integrals are correct by deriving the answers.

Kinematics

A lot of the time, you can find calculus in your physics textbooks. All of the concepts are exactly the same, except for the notations. Here is a table with some of the main terms in physics that are equal to the same terms we have covered.

Conclusion

It is safe to say that calculus can be a tricky subject for many people, but one of the reasons for this is because calculus is conceptual. Once you understand the concept behind calculus, you can apply what you learned to any question. Even though calculus questions can vary, the approach is the same. So the next time you face this subject -or for any other challenging subject- don’t assume you won’t understand it, because sometimes- as shown in this case- it isn’t as hard as you think.