calculus questions!?

Question

In this question, you will estimate the value of the integral

3
S x e^(-x/3)dx
0

using three different approximations.

1. Subdivide the interval [0,3] into three sub-intervals of equal width and complete the following:

Delta x = 1

a0 = 0 f(a0) = ?
a1 = 1 f(a1) = ?
a2 = 2 f(a2) = ?
a3 = 3 f(a3) = ?

x1 = ? f(x1) =?
x2 = ? f(x2) =?
x3 = ? f(x3) =?

2. Calculate the approximate value of the integral using the trapezoidal rule.

(Area) ~=

3. Calculate the approximate value of the integral using the midpoint rule.

(Area) ~=

4. Calculate the approximate value of the integral using Simpson's rule.

(Area) ~=

5. It is possible to show that an antiderivative of x e-x/3 is

-3(x+3) e^(-x/3)

Using this antiderivative, calculate the exact value of the integral.
Integral= ??


Sorry that this is a lot, but any help is appreciated!

Answer 1

All you have to do here is apply formulas. In the 5th exercise all you have to do is apply the fundamental theorem of integral calculus, so that Integral =[ -3(x+3) e^(-x/3)] (0 to 3) = -3(3 + 3)e^(-1) - (-3(0 + 3) e^(-0)] = 3[6 e^(-1) + 3] = 9(2e^(-1) + 1)

Answer 2

This lim is inderterminate (0/0) yet you have precluded using L'Hopital's Rule. So..... e^x = a million + x + x^2/2 + ........ As x is going to 0 then e^x is going to a million + x subsequently your shrink turns into lim (a million + h - a million)/3h = a million/3 lim a million = a million/3 with each and all of the above limits as h has a tendency to 0. bypass in peace....