The Definite Integral

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The Definite Integral

The Monte Carlo Method For Finding Area

Activity

Pick four number from one to 100. Instead of cutting the interval [0,100] into five equal intervals, you can instead cut the interval at the four numbers that you chose and construct five rectangles as before. We will approximate the area of the curve under the curve y =\(\sqrt{x}\) from 1 to 100 to estimate the area.

The Definite Integral

Definition

Let

a = x₀, x₁, ..., x_n = b

be a partition of [a,b] and let

x_i-1 < c_i < x_i

then
\(\sum_{i=1}^n f(c_i)\Delta x_i \)

is called a Reimann Sum, and the limit as the maximum \(\Delta\)x_i approaches 0 is called the definite integral.

Note: f(x) can be negative

Usually to compute a definite integral, we use left or right sums.

Special Cases

\(\int_{a}^{a} f(x)dx = 0 \)

\(\int_{a}^{c} f(x)dx = \int_{a}^{b} f(x)dx + \int_{b}^{c} f(x)dx \)

\(\int_{b}^{a} f(x)dx = -\int_{a}^{b} f(x)dx \)

Example:

\(\int_{-2}^{3} |x| dx = \int_{-2}^{0} -x dx + \int_{0}^{3} x dx \)

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