1. Evaluate the Riemann sum for (x) = x3 − 6x, for 0 ≤ x ≤ 3 with six subintervals, taking the sample points, xi, to be the right endpoint of each interval. Give three decimal places in your answer.

Question

2. Explain, using a graph of f(x), what the Riemann sum in Question #1 represents.

3. Express the given integral as the limit of a Riemann sum but do not evaluate: the integral from 0 to 3 of the quantity x cubed minus 6 times x, dx.

4. Use the Fundamental Theorem to evaluate the integral from 0 to 3 of the quantity x cubed minus 6 times x, dx.

(Your answer must include the antiderivative.)

5. Use a graph of the function to explain the geometric meaning of the value of the integral.

Answer 1

1)

well then the partition is: x={0,0.5,1,1.5,2,2.5,3}

and the step size is h=3/6 = 0.5

x f(x)

0.5 -2.875

1 -5

1.5 -5.625

2 -4

2.5 0.625

3 9

Total -7.875

the approximation is 0.5 * -7.875 = -3.9375

2) it's the area under the curve from x=0 to x=3;

some of it is below the x-axis (negative area) and

some of it is above the x-axis (positive area)

yes they will cancel out, and the integral

gives the exact net result

3) ...... limit of sum of (h * f(xh)) as h-->0

where xh is the right endpoint

4)

(1/4)x^4 - 3x^2 is the anti-derivative

limit as x-->3 : 81/4 - 27 =

81/4 - 108/4=

-27/4 = -6.75

The reason for the discrepency is that the right endpoint

is used to mark off the partitioning..

If the LEFT endpoint is used, the approximation becomes -8.4375

The midpoint rule gives an approximation of -7.03125

The trapezoidal rule gives an approximation of -6.1875

Simpson's 3/8 rule gives an (EXACT) approximation of -6.75

not a surprise