area, left hand?

Question

I'm trying to find the area using left endpoint evaluation but i keep getting the answer wrong!

1) y = x^2 on [ -1,1 ]
n = 8 (number of rectangles)

so I find deltaX = ( 1 - (-1)) / 8 = .25

I now make a table, X values on left, f(x) values on right

Xo = 0 , 0
X1 = .25 , .0625
X2 = .5 , .25
X3 = .75 , .5625
X4 = 1 , 1
X5 = 1.25 , 1.5625
X6 = 1.5 , 2.25
X7 = 1.75 , 3.0625
X8 = 2 , 4

* f(x) values are found by plugging in X into y = x^2

I add f(x0) + f(x1) + .. (fx7) and get 8.75

To find the area, I must multiply 8.75 by deltaX (.25)..I get 2.1875. The answer in the back of my book says 0.6875.

Are my calculations wrong?

i still don't get it..

what do you mean I have it from 0 to 2?

Answer 1

To find out how far off your approximation is, calculate the exact answer by integrating x^2 with respect to x, from -1 to 1...

*integrating*

...which equals 2/3. Looks like your answer is pretty close, given that you're using an 8-point approximation.

Answer 2

Hello

You want it from -1 to 1 and you have it from 0 to 2.

Use (-1,1),(-3/4,9/16),(-1/2,1/4)and
(-1/4,1/16)
and then the rest of yours.

Hope This Helps!!

Answer 3

You made a simple mistake of using values of X from 0 to 2, instead of -1 to 1.

coords at:
x0 = -1, 1
x1 = -0.75, 0.5625
x2 = -0.50, 0.25
x3 = -0.25, 0.0625
x4 = 0,0
x5 = 0.25, 0.0625
x6 = 0.50, 0.25
x7 = 0.75, 0.5625
x8 = 1,1

since we're using left hand margins, we don't use x8

A= .25(x0+x1+x2+x3+x4+x5+x6+x7)
A= .25(1+0.5625+0.25+0.0625+0+0.0625+0.25+0.05625)
A= .25(2.75)
A=0.6875