Showing posts with label area. Show all posts
Showing posts with label area. Show all posts

Saturday, May 4, 2013

Interactive: Mean Value Theorem for Integrals

Mean Value Theorem for Integrals: If \(f\) is continuous on \([a,b]\), then there exists a number \(c\) in \([a,b]\) such that \[\int_a^b {f\left( x \right)dx = f\left( c \right)\left( {b - a} \right)} \]


Instructions: Drag a and b to see c calculated dynamically. The area under the curve from a to b is the same as the area of the pink rectangle formed by b - a and f(c).



Interactive: Net Area

Net Area: If the function f takes on both positive and negative values, then the Riemann sum is the result of the area under the curve above the x-axis less the area under the x-axis above the curve.


Instructions: Drag a and b to see net area change sign. Purple indicates positive net area and pink indicates negative net area.



Interactive: Definite Integral

Definite Integral: If \(f\) is a continuous function defined on the interval \([a, b]\) that is divided into \(n\) equal subintervals \(\Delta x = \frac{{b - a}}{n}\) where \(x_i^*\) lies in the ith subinterval then the definite integral of \(f\) from \(a\) to \(b\) follows:

\[\int_a^b {f\left( x \right)dx} = \mathop {\lim }\limits_{n \to \infty } \sum\limits_{i = 1}^n {f\left( {x_i^*} \right)} \,\Delta x\]


Instructions: Drag the slider to see that the area under the curve is the limit of Riemann sums. You can also change the sums to left, middle, right, and trapezoidal below the graph.

Riemann Sum: Left Middle Right Trapezoidal