# B Integral test and its conclusion

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1. Aug 3, 2016

### The Subject

When you take the integral of f(x) and gives you some value. What are you supposed to conclude from this value?

2. Aug 3, 2016

### Stephen Tashi

Look at an example. What value do you get from $$\int_0^\infty 3x\ dx$$ ?

3. Aug 3, 2016

### The Subject

ok so I get
$$\lim_{t \to \infty} \int_0^t 3x dx = \lim_{t \to \infty} \frac{3}{2}x^2 |_0^t=lim_{t \to \infty} \bigg(\frac{3}{2}t^2 - \frac{3}{2}0^2\bigg)=\infty$$

4. Aug 3, 2016

### Stephen Tashi

So what does the integral test say about the convergence or divergence of the infinite series $0 + 3 + 6 + 9 + 12 + ...$ ?

5. Aug 3, 2016

### The Subject

The series 0 + 3 + ... diverges. Since f(x) div, an also diverges. I get it how to use it now.

Thanks!

6. Aug 3, 2016

### Stephen Tashi

Sketch a function with a positive graph and, on top of that, sketch the rectangles whose areas represent the terms of the related series. These rectangles have bases [0,1], [1,2] ... etc. and heights determined by the function's value at the left endpoints. The area of the rectangles is not a particularly good approximation to the area under the graph, but the intuitive idea is that the two areas are either both finite or both infinite.