Dido525 2 years ago Definition of the definite integral question:

1. Dido525

Using the definition of the definite integral, compute: $\int\limits_{-4}^{2}(3x^2+12x+20)dx$

2. malical

Is the definition of a definition integral FTC or the definition of an integral: lim (x,y)-> (0,0) and on and on?

3. jayz657

take the anti derivative of 3x^2 + 12x + 20 is x^3 +6x + 20x then you plug in 2 and then -4 (2)^3 + 6(2) + 20(2) - [ -4^3 + 12(-4) + 20(-4) ] 8 + 12 + 40 - [ -64 - 48 - 80 ] 60 - [ - 192] = 252

4. Dido525

Definition of integral. @jayz657 Thanks but I want an explanation on this concept.

5. Dido525

I can calculate the anti derivative fine but I have to use Riemann sums @jayz657 .

6. Dido525

Just an explanation of some sort for Riemann sums would be fine. I can do the rest I am sure.

7. jayz657

|dw:1353040377471:dw| this is the riemann sum, you are drawing an infinite amount of rectangles under the curve here you know the length is change of x and the height is f(x) and you sum up each rectangle to get the area under the curve so will will get this $\sum_{-4}^{2} f(x)\Delta x$

8. Dido525

Yes I know the fundamental concept. But how would I exactly apply it?

9. jayz657

when you sum up the infinite amount of rectangles you will get the integral there

10. malical

You integrate the equation and then you get a function. That function will be in terms of f(x). The top number in the definite integral is b and the lower is a. f(b)-f(a) is your answer. Jayz got 252

11. Dido525

Yeah but How would I exactly sum up an infinite number of rectangles for my given function?

12. malical

Do you want to see how to do this problem or the theory behind it?

13. Dido525

@malical : I know how to find an antiderivative. I just have trouble applying riemann sums.

14. malical

Why are Riemann sums important?

15. Dido525

The question specifically says to use Riemann sums.

16. Dido525

Never mind. I got it :) .

17. jayz657

the change of x is the rate of change of each rectangle and rate of change is related to the derivative, dx so in this equation you can just draw as many rectangles as you want under the curve using a delta x width and using f(x) as your height heres an exmaple doing it the long way lets say i make 4 rectangles with length 3 |dw:1353040955063:dw|

18. Dido525

Yeah I got it thanks :) .

19. jayz657

ok np