Welcome to the Applications of Integration!
Hello everyone! Let's start our journey into one of the highlights of Mathematics III: "Applications of Integration."
You might be thinking, "Calculus looks like it's going to be a chore to calculate..." But the true nature of integration is actually a very simple concept: "breaking something into small pieces and adding them all up."
In this chapter, we will learn how to use integration to find areas, volumes, and the lengths of curves. Shapes that look difficult at first glance can be calculated almost like magic using the power of integration. Don't rush; let’s master this one step at a time!
1. Area
First, let's cover the basics of area. While you’ve learned about areas in Mathematics II, in Mathematics III, we will handle more complex functions (fractional functions, logarithmic functions, trigonometric functions, etc.).
1-1. Area Between Two Curves
The area \( S \) enclosed by two curves \( y = f(x) \) and \( y = g(x) \) can be found using the following formula:
\( S = \int_{a}^{b} \{ f(x) - g(x) \} dx \)
(Provided that \( f(x) \geqq g(x) \) on the interval \( [a, b] \))
【Pro Tip: Top minus Bottom!】
When finding the area, remember that you should always integrate "(upper graph) - (lower graph)." If you're not sure which is on top, the trick is to plug in a test value or sketch a quick graph to see the orientation.
1-2. Integration with Respect to the \( y \)-axis
Depending on the graph, it is sometimes easier to think in terms of the \( y \)-axis rather than the \( x \)-axis.
The area \( S \) enclosed by a curve \( x = g(y) \), the \( y \)-axis, and lines \( y = c, y = d \) is:
\( S = \int_{c}^{d} g(y) dy \)
In this case, we think in terms of "(right graph) - (left graph)."
【Common Mistake】
If you reverse the upper and lower limits of integration, your area will come out negative. Remember that "area must always be positive!" If your calculation results in a negative value, it’s a sign that you have the order of the graphs flipped.
Summary: The Secrets of Area
・Basically, use "Top minus Bottom" or "Right minus Left."
・Always sketch a simple graph to check the top/bottom (or left/right) relationship!
2. Volume
Volume becomes much easier to understand if you imagine "stacking thin slices (cross-sections)."
2-1. Volume from Cross-Sectional Area
If the cross-sectional area of a solid when cut by a plane perpendicular to the \( x \)-axis is \( S(x) \), then the volume \( V \) of that solid is:
\( V = \int_{a}^{b} S(x) dx \)
【A Familiar Analogy】
Think of slicing a cucumber. The total volume of the cucumber is just the sum of the areas of each individual slice from one end to the other, right? That is the exact concept of integration!
2-2. Volume of a Solid of Revolution (around the \( x \)-axis)
The volume \( V \) of a solid formed by rotating the area enclosed by the curve \( y = f(x) \) and the \( x \)-axis around the \( x \)-axis is:
\( V = \pi \int_{a}^{b} \{ f(x) \}^2 dx \)
The reason for this formula is that each cross-section is a "circle with radius \( f(x) \)." Since the area of a circle is \( \pi \times (\text{radius})^2 \), this naturally follows.
2-3. Volume of a Solid of Revolution (around the \( y \)-axis)
If you rotate a curve \( x = g(y) \) around the \( y \)-axis, the volume is:
\( V = \pi \int_{c}^{d} \{ g(y) \}^2 dy \)
In this case, the radius of the circular cross-section is the \( x \)-coordinate (which is \( g(y) \)).
【Fun Fact】
Think of a Baumkuchen cake. That is also a type of solid of revolution. The way the thin layers are stacked from the center outward is the perfect visual representation of integration.
Summary: The Secrets of Volume
・The basic step is to "find the cross-sectional area first, then integrate."
・For solids of revolution, remember that it takes the form \( \pi \int (\text{radius})^2 \).
3. Length of a Curve
You can even use integration to calculate the "length of curved lines" that a straight ruler cannot measure.
3-1. Length of a Parametric Curve
The length \( L \) of a curve represented by \( x = f(t), y = g(t) \) as \( t \) goes from \( \alpha \) to \( \beta \) is:
\( L = \int_{\alpha}^{\beta} \sqrt{ \left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2 } dt \)
3-2. Length of a Graph \( y = f(x) \)
The length \( L \) of a function \( y = f(x) \) as \( x \) goes from \( a \) to \( b \) is:
\( L = \int_{a}^{b} \sqrt{ 1 + \{ f'(x) \}^2 } dx \)
【Study Tip: The Pythagorean Theorem】
This complex square root formula actually comes from the Pythagorean Theorem \( a^2 + b^2 = c^2 \). We consider the "horizontal change" and "vertical change" in an infinitesimal interval as the two sides of a right triangle, and we are finding the hypotenuse (the infinitesimal length of the curve). Doesn't that make it feel a bit more approachable?
Summary: The Secrets of Curve Length
・When calculating the contents of the square root, it often takes the form \( ( \dots )^2 \), which allows the root to be removed.
・Calculations can get messy, so watch out for differentiation errors!
4. Velocity and Distance
While this topic appears in Physics class, we will handle it in more detail using the integration from Mathematics III.
4-1. Motion on a Straight Line
Let \( v(t) \) be the velocity at time \( t \), and \( x_0 \) be the position at time \( a \).
・Position at time \( t = b \): \( x(b) = x_0 + \int_{a}^{b} v(t) dt \)
・Displacement (change in position) from time \( a \) to \( b \): \( \int_{a}^{b} v(t) dt \)
・Total distance traveled from time \( a \) to \( b \): \( \int_{a}^{b} | v(t) | dt \)
【Important: Displacement vs. Total Distance】
The difference between these two is very important!
For example, if you "go 5 meters to the right and 3 meters back to the left":
・The displacement is \( 5 - 3 = 2 \) meters.
・The total distance is \( 5 + 3 = 8 \) meters.
When finding total distance, you must use the absolute value to ensure the "distance back" is also counted as positive.
4-2. Motion on a Plane (Distance)
If the velocity components of a point moving on a plane at time \( t \) are \( (v_x, v_y) \), the total distance \( S \) is:
\( S = \int_{\alpha}^{\beta} \sqrt{ (v_x)^2 + (v_y)^2 } dt \)
This has the exact same form as the formula for "length of a curve!" You are simply integrating the magnitude of the velocity over time to find the total distance traveled.
Summary: The Secrets of Velocity and Distance
・Read the problem carefully to see if it's asking for "position" or "total distance."
・For total distance, don't forget the absolute value (or the Pythagorean form).
Final Thoughts
The best way to master the applications of integration is not to memorize the formulas by heart, but to visualize "what the goal of this calculation is (Area? Volume? Length?)".
The calculations might feel heavy and difficult at first, but with practice, you will definitely be able to solve them fluently. Start with the textbook examples, and keep moving forward one step at a time!
I'm rooting for you!