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

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

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

To approximate the integral using 5 inscribed rectangles, you can use the formula for the Riemann sum:

Riemann Sum = ∑(f(xi) * Δx)

where:
- f(xi) represents the function evaluated at each x value,
- Δx is the width of each rectangle, and
- ∑ denotes the sum over all the rectangles.

Using your given x values (x1 = 1.4, x2 = 1.8, x3 = 2.2, x4 = 2.6, and x5 = 3.0), we can calculate the width of each rectangle:

Δx = (b - a) / n

where:
- n is the number of rectangles,
- a is the lower limit, and
- b is the upper limit.

In this case, a = 1 and b = 3. Since there are 5 rectangles, we have:

Δx = (3 - 1) / 5 = 0.4

Now, we can calculate the Riemann sum using the given formula:

Riemann Sum = f(x1) * Δx + f(x2) * Δx + f(x3) * Δx + f(x4) * Δx + f(x5) * Δx

To evaluate f(xi), substitute xi into the function (1/x) and multiply it by Δx. Then perform this calculation for each value of xi and sum them all up.

Riemann Sum = (1/1.4) * 0.4 + (1/1.8) * 0.4 + (1/2.2) * 0.4 + (1/2.6) * 0.4 + (1/3.0) * 0.4

Simplifying this expression will give you an approximation of the definite integral of (1/x) from 1 to 3 using 5 inscribed rectangles.

To approximate the integral using the Riemann Sums method with 5 inscribed rectangles, you need to calculate the area of each rectangle and then sum them up.

The width of each rectangle can be found by taking the difference between the x-values of consecutive rectangles, which in this case is 1.4 - 1 = 0.4.

To calculate the height of each rectangle, you need to evaluate the function (1/x) at each rectangle's x-value. So, for rectangle 1, the height (h1) is given by evaluating (1/x) at x1, which is 1/1.4. Similarly, you can find h2, h3, h4, and h5 using the respective x-values.

Once you have the width and height of each rectangle, you can find the area of each rectangle by multiplying the width by the height. So, the area of rectangle 1 (A1) is (0.4) * (1/1.4). Similarly, calculate A2, A3, A4, and A5.

Finally, to approximate the integral, you sum up the areas of all the rectangles: A1 + A2 + A3 + A4 + A5.

In this case, with the given x-values, you can substitute them into the appropriate formulas to calculate the areas and then sum them up to get the approximation of the integral.