evaluate integral from 2 to 0 on f(x)dx?

Question

Evaluate (integral from 2 to 0) f(x)dx,
where f(x) = 3 x^{9}, & (0 less than or equal to x < 1)
2 x^{4}, & (1 less than or equal to x less than or equal to 2)

Answer 1

Well, the integral of x^9 is 1/10 x^10. Differentiate the latter to see why. The integral of x^4 is 1/5 x^5; again, differentiate to see why.

The good thing about such simple formulas for integration is that if you're not sure you remembered them correctly, it's easy to check them (by differentiation) and see if you got it right.

If you can integrate x^9, you know how to integrate 3x^9, right? And the same goes for a constant multiple of anything else you can integrate.

So for the first one, just evaluate 3/10 x^10 at 2 and at 0, and subtract the second value from the first one.

For the second problem, similarly, find 2/5 2^5 - 2/5 1^5.