Welcome to Chapter 6: Applications of Integration! Now that we have mastered the fundamental techniques of integration, we are ready to apply them to geometry. In tonight's class, we begin with Section 6-1: Areas Between Curves.

Previously, we used the definite integral to find the area under a single curve (between the curve and the $x$-axis). Now, we are expanding that concept to find the area of a region sandwiched between two different functions, $f(x)$ and $g(x)$.

1. The Standard Vertical Approach (dx)

When measuring the area between two curves, we look at the height of a representative rectangle slice. If $f(x)$ is the upper curve and $g(x)$ is the lower curve, the height is $f(x) - g(x)$. By summing these rectangles as the width approaches zero, we get our formula:

$$ A = \int_{a}^{b} [f(x) - g(x)] \, dx $$

Key Condition: This formula applies when $f(x) \ge g(x)$ for all $x$ in the interval $[a, b]$.

2. Dealing with Intersecting Curves

What happens if the curves cross each other? As seen in the attached notes (specifically the sine and cosine example), the function on "top" changes.

  • Step 1: Find the points of intersection by setting $f(x) = g(x)$. These become your limits of integration.
  • Step 2: Split the integral into separate regions. You must calculate the area for each section where the top/bottom orientation stays the same and add them together.

Technically, the area is defined as the integral of the absolute value of the difference:

$$ A = \int_{a}^{b} |f(x) - g(x)| \, dx $$

3. Integrating with Respect to y (Horizontal Slices)

Sometimes, solving for $y$ in terms of $x$ is difficult, or the curves are shaped in a way that makes vertical slicing complicated. In these cases, it is often easier to treat $x$ as a function of $y$ (i.e., $x = f(y)$ and $x = g(y)$).

Instead of "Top minus Bottom," we use "Right minus Left":

$$ A = \int_{c}^{d} [x_R - x_L] \, dy $$

where $x_R$ is the right-hand boundary curve and $x_L$ is the left-hand boundary curve.

Tips for Success

  • Sketch the graph: It is nearly impossible to set up these integrals correctly without a visual representation of which curve is on top (or on the right).
  • Watch your signs: Remember that area must always be positive. If you get a negative answer, check if you swapped your functions or missed an intersection point.
  • Review the Examples: The attached PDF walks through Example 1 ($e^x$ and $y=x$) and Example 6 (Trig functions). Make sure to review these step-by-step solutions.

Download the class notes below to follow along with the examples!