Area: Measurement, Formulas, Calculus, and Optimization

All flashcards

What is area?

Area is a measure of the two-dimensional size of a region on a surface.

How does surface area differ from plane area?

Plane area measures a flat region, while surface area measures the boundary surface of a three-dimensional object.

What is the rectangle rule for area?

A rectangle with length $h$ and breadth $k$ has area $a(R)=hk$.

What does congruence invariance mean for area?

If two figures are congruent, they have the same area, regardless of their position or orientation.

How can step regions help define area?

Finite collections of rectangles can approximate a region from below and above; when the approximations determine a unique value, that value is the region's area.

Quiz questions

1. Why is the overlap subtracted in the area rule for a union?Answer: Because the overlap is counted once in each individual area and would otherwise be counted twice.
2. How many square inches are in one square foot?Answer: 144 square inches.
3. For the triangle formula $A=\frac{1}{2}bh$, how is height $h$ measured relative to base $b$?Answer: As the perpendicular distance from the base to the opposite vertex.
4. Why does the shoelace formula use the absolute value of its coordinate sum?Answer: The sum's sign depends on the order in which the vertices are listed.
5. If $f(x)\geq g(x)$ on $[a,b]$, which integral gives the area between their graphs?Answer: $\int_a^b (f(x)-g(x))\,dx$