Area under the curve

Definition

The area under the curve is a measure of the region bounded by a given function, the x-axis, and vertical lines at two specified points. It is often computed using definite integrals.

5 Must Know Facts For Your Next Test

1. The area under the curve from $a$ to $b$ is found using the definite integral: $$\int_{a}^{b} f(x) \, dx$$.
2. Approximations of the area under a curve can be done using Riemann sums, trapezoidal rule, or Simpson's rule.
3. A positive value for $f(x)$ over an interval results in a positive area, while a negative value results in a negative area.
4. The Fundamental Theorem of Calculus connects differentiation and integration, showing that the definite integral can be evaluated using antiderivatives.
5. Partitioning an interval into smaller subintervals increases the accuracy of approximations for the area under a curve.

Review Questions

• How do you calculate the exact area under a curve between two points?
• What are some common methods to approximate the area under a curve?
• Explain how the Fundamental Theorem of Calculus relates to finding areas under curves.

Related terms

Definite Integral: A type of integral that calculates the net area between two points on a curve, denoted as $$\int_{a}^{b} f(x) \, dx$$.

Riemann Sum: An approximation method for calculating the area under a curve by dividing it into small rectangles or trapezoids.

Fundamental Theorem of Calculus:

A theorem stating that differentiation and integration are inverse operations; it provides a way to evaluate definite integrals using antiderivatives.