integrals pls help asap

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January 21st 2008, 06:11 PM
memena

integrals pls help asap

Estimate the area under the graph of f(x) = e^-x^2 from x = -2 to x = 2 using four approximating rectangles and midpoints.

(Doh):confused:
January 21st 2008, 06:18 PM
Jhevon

Quote:

Originally Posted by memena

Estimate the area under the graph of f(x) = e^-x^2 from x = -2 to x = 2 using four approximating rectangles and midpoints.

(Doh):confused:

divide the interval [-2,2] into four equal sub-intervals. that is, we partition it into the intervals: [-2,-1], [-1,0], [0,1] and [1,2]

now, take the midpoints of these rectangles, and label them by $x_i$ for $i = 0,1,2,...$ . so we have $x_0 = -1.5, ~x_1 = -0.5, ~x_2 = 0.5, \mbox{ and } x_3 = 1.5$

now we can use a finite Riemann sum to estimate the area. thus, an estimate of the area is given by:

$A = \sum_{i = 0}^3 f(x_i) \Delta x$

where $\Delta x = x_1 - x_0 = x_2 - x_1 = ... = 1$ and of course, $f(x) = e^{-x^2}$
January 21st 2008, 06:36 PM
Jhevon

to expound:

since $\Delta x = 1$ , the area estimate is given by:

$\sum_{i = 0}^3 f(x_i) = f(x_0) + f(x_1) + f(x_2) + f(x_3)$

now i hope you can continue from there

if you are not clear on something, say so
January 21st 2008, 07:02 PM
memena

ummm

ok do i draw this graph and then add the rectangles, if so can u tell me what it is supposed to look like, im sorry i am completely lost, i don't understand anything!(Speechless)
January 21st 2008, 07:12 PM
mr fantastic

Quote:

Originally Posted by memena

ok do i draw this graph Mr F says: Yes, use a graphics calculator.

and then add the rectangles, Mr F says: Yes, add the area of each of the four rectangles. As has already been stated, the total area is $f(x_0) + f(x_1) + f(x_2) + f(x_3)$ , where $x_0 = -1.5, ~x_1 = -0.5, ~x_2 = 0.5 \,$ and $\, x_3 = 1.5$ .

if so can u tell me what it is supposed to look like, Mr F says: Draw the graph (use a graphics calculator to get it) and draw in the rectangles. Then you'll know what it looks like.

im sorry i am completely lost, i don't understand anything!(Speechless) Mr F says: But you must surely have some examples to follow ....?

..
January 21st 2008, 07:14 PM
Jhevon

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Quote:

Originally Posted by memena

ok do i draw this graph and then add the rectangles, if so can u tell me what it is supposed to look like, im sorry i am completely lost, i don't understand anything!(Speechless)

yes, you can draw a diagram to visualize it.

here:

basically, your finding the area of the 4 rectangles we have in the diagram, that is, length times width for each. the height of each rectangle is given by $f(x_i)$ , the widths are all one