Unit 8: Applications of Integration — Area Between Two Curves

Net Signed Area

The definite integral $\int_a^b f(x) \, dx$ represents the total area between the curve and the x-axis, considering above and below the axis.

Area Between Curves

The total geometric area of the region bounded by two functions, calculated as a definite integral, and is always positive.

Vertical Slicing

A method where rectangles are oriented upright, with differential width $dx$.

Horizontal Slicing

A method where rectangles are oriented sideways, with differential width $dy$.

Area Formula (Vertical Slicing)

Area = $\int_{a}^{b} [f(x) - g(x)] \, dx$, where $f(x)$ is the upper function and $g(x)$ is the lower function.

Top Minus Bottom

A memory aid for vertical slicing, indicating that to find area, subtract the lower function from the upper function.

Intersection Points

The points where two functions meet, found by solving $f(x) = g(x)$.

Fundamental Theorem of Calculus

A principle that connects differentiation with integration, allowing the evaluation of definite integrals.

Complex Regions

Regions where curves intersect, necessitating splitting the integral into sub-regions based on the areas bounded by each function.

Split Integral Formula

Area = $\int{a}^{c} [Top1 - Bottom1] \, dx + \int{c}^{b} [Top2 - Bottom2] \, dx$ for intersecting curves.

Horizontal Slicing Area Formula

Area = $\int_{c}^{d} [f(y) - g(y)] \, dy$, with $f(y)$ as right and $g(y)$ as left functions.

Right Minus Left

A memory aid when slicing horizontally, signifying the area is found through the difference of right and left boundaries.

Common Mistake: Mismatched Limits

Error of using $x$-limits while integrating with respect to $dy$; limits should be $y$-values.

Common Mistake: Order of Subtraction

Error of subtracting Bottom - Top; area must be positive, thus Top - Bottom is correct.

Common Mistake: Intersection Points as Limits

Assuming integration always stops at intersection points; intersection points define a region but not the limits.

Common Mistake: Forgetting to Split

Not splitting the integral when curves cross within the interval, which is necessary to correctly calculate area.

Vertical vs Horizontal Slicing

Vertical uses $dx$ with Top and Bottom functions; Horizontal uses $dy$ with Right and Left functions.

Area Enclosed by Curves Example - Basic Vertical Slicing

Example shows how to find the area between $y=x^2+2$ and $y=x$ from $x=[0, 2]$.

Area Enclosed by Intersecting Curves

Example of finding area between $f(x) = \sin(x)$ and $g(x) = \cos(x)$ with an intersection at $x = \pi/4$.

Area Between Parabola and Line Example

Calculating the area between $x = y^2$ and line $x = y + 2$ through intersection points.

Area Calculation Step 1: Sketch Graph

First step to finding area between curves; identify which function is higher or lower.

Area Calculation Step 2: Identify Top and Bottom

Identify which function is above and which is below on the specified interval.

Area Calculation Step 3: Set Up Integral

Set up the integral by subtracting the lower function from the upper function before integrating.

Area Calculation Step 4: Evaluate Integral

Use the results from the integral to calculate the total area.

Comparison Table: Features of x vs y

Table outlines differences between functions of x (vertical) and functions of y (horizontal) regarding slicing and integration.