Estimate definite intergral with 4 partitions...?

Question

Estimate definite intergral of ∫√x dx, a = 0, b = 1, using lower sum with 4 partitions.

Not sure how to approach this problem...

Answer 1

See my diagram: http://i6.tinypic.com/6tns5xt.gif

The definite integral (which equals the area under the sqrt curve in red) is to be approximated by the rectangles outlined in blue. (The first rectangle is of zero height.)

Note that the interval [0,1] has been divided into four parts. Since √x is an increasing function, the value for the lower sum is the function value at the left endpoint of each partition. These are the "heights" of the rectangles. The width of each rectangle is ¼ (the length of the interval divided by the number of partitions).

The area of the four rectangles is

¼*√0 + ¼*√0.25 + ¼*√0.5 + ¼*√0.75, or

¼ (√0 + √0.25 + √0.5 + √0.75)

I'll let you do the arithmetic. You should get around 0.518. We know that the correct value is 2/3, so it's a pretty rough estimate. This is not surprising, since (as you can see from my diagram) a lot of the area under the curve gets left out with such a coarse partitioning.