Areas Between Curves: Calculus Study Guide

Learning Objectives

By the end of this study guide, you should be able to:

• Determine the area of a region between two curves by integrating with respect to the independent variable ($x$).
• Find the area of a compound region by breaking it into separate integrals where the bounding curves intersect and switch positions.
• Determine the area of a region between two curves by integrating with respect to the dependent variable ($y$) using horizontal slicing.

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Key Terms & Glossary

• Compound Region: A complex area where the bounding functions intersect within the interval, requiring the area to be split into multiple integrals. Example: Finding the area between the crossing paths of a jet plane and a drone to determine potential collision zones.
• Representative Rectangle: A geometric tool used to approximate an infinitesimally thin slice of area between curves. Example: Thinking of a curved plot of land as being divided by perfectly straight, thin fences stacked next to each other.
• Limits of Integration: The geometric boundaries ($x=a$ to $x=b$, or $y=c$ to $y=d$) that define the start and end of the calculated area. Example: The exact start and end times (bounds) when analyzing the accumulated difference between energy produced and energy consumed in a solar grid.

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The "Big Idea"

In early calculus, you learned to find the area under a single curve (between the curve and the $x$-axis). The "Big Idea" here is that we can expand this concept to find the exact area trapped between any two curves.

Instead of integrating just $f(x), you integrate the difference between the two functions: (Top - Bottom)$ or $(Right - Left)$. This fundamental principle allows engineers, physicists, and economists to calculate bounded regions—such as the exact physical material needed to fill a mold, or the total profit margin between revenue and cost curves over time.

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Formula / Concept Box

[!IMPORTANT] Always remember that Area must be positive. If you get a negative result, you likely subtracted the larger function from the smaller one!

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Hierarchical Outline

1. Introduction to Areas Between Curves
   • Expanding definite integrals beyond the $x$-axis.
   • Approximating with Representative Rectangles.
2. Regions Defined with Respect to $x$ (Vertical Slicing)
   • Identifying the top curve $f(x)$ and bottom curve $g(x)$.
   • Setting the upper and lower Limits of Integration ($a$ and $b$).
3. Compound Regions and Intersecting Graphs
   • Finding intersection points algebraically.
   • Splitting the primary integral into multiple distinct integrals.
4. Regions Defined with Respect to $y$ (Horizontal Slicing)
   • Re-expressing functions as $x = u(y)$ and $x = v(y)$.
   • Simplifying integrals when functions cross multiple times vertically but not horizontally.

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Visual Anchors

1. Decision Matrix Flowchart

2. Geometric Representation of Area between Curves

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Definition-Example Pairs

• Integrating with respect to $x$ ($dx$)

  • Definition: Slicing the area vertically into infinite rectangles of width $dx, where the height is the y$-value difference.
  • Example: Calculating the 2D cross-sectional area of an airplane wing by measuring the difference between the upper contour and lower contour at various points along its length.

• Integrating with respect to $y$ ($dy$)

  • Definition: Slicing the area horizontally into infinite rectangles of height $dy, where the width is the x$-value difference.
  • Example: Determining the fluid capacity of an irregular vase by summing thin horizontal discs of water from the base to the lip.

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Comparison Tables

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Worked Examples

▶Example 1: Finding Area with Vertical Slices (Compound Region)

Problem: Find the area bounded by $y = x^3$ and $y = x$ on the interval $[-1, 1]$.

Step 1: Find points of intersection. Set the equations equal to each other: $x^3 = x$ $x^3 - x = 0$ $x(x^2 - 1) = 0 \implies x = -1, 0, 1$

Step 2: Determine Top and Bottom curves for each interval.

• On $[-1, 0]$: Test $x = -0.5$. $y_1 = (-0.5)^3 = -0.125$, and $y_2 = -0.5$. Since $-0.125 > -0.5$, $x^3$ is the Top curve.
• On $[0, 1]$: Test $x = 0.5$. $y_1 = (0.5)^3 = 0.125$, and $y_2 = 0.5$. Since $0.5 > 0.125$,$x$ is the Top curve.

Step 3: Set up and evaluate the compound integral. $A = \int_{-1}^{0} (x^3 - x) \,dx + \int_{0}^{1} (x - x^3) \,dx$

Find the antiderivatives: $A = \left[ \frac{x^4}{4} - \frac{x^2}{2} \right]_{-1}^{0} + \left[ \frac{x^2}{2} - \frac{x^4}{4} \right]_{0}^{1}$

Evaluate at the bounds: $A = \left( 0 - \left[\frac{1}{4} - \frac{1}{2}\right] \right) + \left( \left[\frac{1}{2} - \frac{1}{4}\right] - 0 \right)$ $A = \left( 0 - \left[-\frac{1}{4}\right] \right) + \left( \frac{1}{4} \right) = \frac{1}{4} + \frac{1}{4} = \frac{1}{2} \text{ units}^2$

▶Example 2: Finding Area with Horizontal Slices (Integrating w.r.t $y$)

Problem: Find the area bounded by $x = y^2$ and $x = 2 - y^2$.

[!TIP] Because the equations are already given in terms of $y$ (e.g., $x = f(y)), it is highly efficient to integrate with respect to y$ using horizontal slices!

Step 1: Find points of intersection. Set the equations equal to each other to find $y$-bounds: $y^2 = 2 - y^2$ $2y^2 = 2 \implies y^2 = 1 \implies y = -1, y = 1$

Step 2: Determine Right and Left curves. Test a $y$-value between $-1$ and 1, such as $y=0$.

• Left curve: $x = (0)^2 = 0$
• Right curve: $x = 2 - (0)^2 = 2$ The right curve is $x = 2 - y^2$.

Step 3: Set up and evaluate the integral. $A = \int_{-1}^{1} [(2 - y^2) - (y^2)] \,dy$ $A = \int_{-1}^{1} (2 - 2y^2) \,dy$

Find the antiderivative: $A = \left[ 2y - \frac{2y^3}{3} \right]_{-1}^{1}$

Evaluate at the bounds: $A = \left( 2(1) - \frac{2(1)^3}{3} \right) - \left( 2(-1) - \frac{2(-1)^3}{3} \right)$ $A = \left( 2 - \frac{2}{3} \right) - \left( -2 + \frac{2}{3} \right)$ $A = \left( \frac{4}{3} \right) - \left( -\frac{4}{3} \right) = \frac{8}{3} \text{ units}^2$

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Checkpoint Questions

1. What geometric shape is fundamentally used to approximate the exact area between two curves before taking the limit? Answer: The rectangle. We use infinitely many, infinitesimally thin "representative rectangles" to sum up the total area.

2. If $f(x)$ and $g(x) cross each other twice inside the boundary interval [a, b]$, how many definite integrals will you need to write to calculate the total bounded area? Answer: Three. The interval must be split at both intersection points, creating three distinct zones where the "Top" and "Bottom" curves swap.

3. You are looking at a graph bounded by $y = \sqrt{x}$ and $y = x - 2, but finding the area with vertical slices (dx$) requires splitting the region because the bottom boundary changes partway through. What is the alternative strategy? Answer: Integrate with respect to $y (horizontal slicing). By rearranging the functions to x = y^2$ and $x = y + 2$, the "Right" and "Left" bounds remain perfectly consistent over the whole region, requiring only one integral.

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Muddy Points & Cross-Refs

• Confusing Bounds: A common mistake is using $x-values for the bounds when integrating with respect to y. If your integrand has a dy, your limits of integration **must** be y$-values.
• Absolute Value connection: Conceptually, you are integrating $$\int$|f(x) - g(x)| dx$. Setting up "Top minus Bottom" is the geometric way of evaluating that absolute value.
• Further Study: This concept directly bridges into "Volumes by Slicing" (the Disk/Washer methods). Mastery of determining Top vs Bottom / Right vs Left curves is crucial for determining radiuses in volume calculations.