Area under the curve

Definition

The area under the curve represents the integral of a function over a given interval on a graph. It quantifies the accumulated value between the curve and the x-axis.

5 Must Know Facts For Your Next Test

1. The area under the curve can be approximated using methods such as Riemann sums, trapezoidal rule, and Simpson's rule.
2. The definite integral from $a$ to $b$ of a function $f(x)$ gives the exact area under the curve between these two points.
3. $\int_a^b f(x) \, dx$ is used to denote the definite integral which calculates this area.
4. If $f(x)$ is above the x-axis, the area is positive; if below, it is negative.
5. Understanding geometric interpretation helps in visualizing problems related to work done by forces or total distance traveled.

Review Questions

• What methods can be used to approximate the area under a curve?
• How do you denote the definite integral that represents the area under a curve from $a$ to $b$?
• What does it mean when an integral yields a negative area?

Related terms

Definite Integral:

A type of integral that calculates the net area under a curve over an interval $[a, b]$.

Riemann Sum:

A method for approximating the total area under a curve by summing up areas of multiple rectangles.

Trapezoidal Rule:

\text{An approximation method for calculating integrals by dividing} \ [a,b]\ \text{into intervals and estimating areas using trapezoids.}