Question

Use the Riemann Sums corresponding to 5 inscribed rectangles of equal width to approximate the integral a= 1, b= 3, (1/x)dx

this is all for definite integral

i just know x1=1.4, x2=1.8, x3=2.2, x4=2.6, x5=3.0

how do i continue

Answer 1

you're halfway there. We approximate the integral by adding up the areas of the rectangles. Since the width of each rectangle is 0.4, all we have to do is evaluate f(x) at some point in each interval and multiply.

Usually the left side, middle, or right side of the rectangle is chosen, but as the actual value of the integral is the limit as the width decreases to zero, it doesn't really matter where in the interval f(x) is evaluated.

Since you have specified inscribed rectangles, and since 1/x is concave upward for x>0, we will want the right end of each interval. (If that's not clear, sketch the curve and see where the rectangles intersect the curve.)

So, just add up

0.4 (f(1.4)+f(1.8)+...+f(3))
0.4 (.714+.556+.455+.385+.333)
0.4 * 2.442
= 0.977

Since our rectangles are all under the curve, our estimate will be low.

Actual value: 1.0986

Answer 2

wow I didn't think it was so simple. thank you !