Engineering Class 11 – Integration

Hi,

Integration is the concept of successive summation in order to calculate the area under a curve.

In class we solved this problem:

Compute the integral $\int_{0}^{4}x^3 dx$ by computing the Riemann sums for a regular partition.

Using Mathematica we obtained the following:

$\Delta x = \frac{b-a}{n} = \frac{4-0}{n} = \frac{4}{n}$

$X_i = a + i \Delta x = \frac{4 i}{n}$ , for each i

$f(X_i) = \left (\frac{4 i}{n} \right )^3 = 64 \frac{i^3}{n^3}$, for each i

$\sum_{i=1}^{n} f(x_i)\Delta x = \sum_{i=1}^{n} 64 \frac{i^3}{n^3}\frac{4}{n} = \frac{256}{n^4} \frac{n^2 (n+1)^2}{4}$

$\int_{0}^{4} x^3dx= \lim_{n \to \infty } \sum_{i=1}^{n} 64 \frac{i^3}{n^3}\frac{4}{n}$

$= \lim_{n \to \infty } \frac{256}{n^4} \frac{n^2 (n+1)^2}{4} = 64$

Please note that for a very large n, $n^2(n+1)^2 = n^4$

$\lim_{n \to \infty } \frac{256}{n^4} \frac{n^2 (n+1)^2}{4} = \lim_{n \to \infty } \frac{256}{n^4} \frac{n^4}{4} = \frac{256}{4} = 64$

Now the easy way:

We then used Mathematica to get the same answer : …much easier :)

-Edward

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