Given the table below for selected values of f(x), use 6 right rectangles to estimate the value of the integral from 1 to 10 of f(x)dx.

Table:
1 3 4 6 7 9 10
4 8 6 10 10 12 16

The only problem I'm having with this is figuring out the values of f(2.5), f(5.5), and f(8.5) since each rectangle should be 1.5 units apart. Is there a way to solve this without those values?

Steve today at 10:43pm
the area is

6
∑ f(xk) ∆xk
k=1

where the xk are 3,4,6,7,9,10
and the ∆xk are 2,1,2,1,2,1

so just add 'em up.

I don’t understand what Steve is explaining

the problem involves a Riemann sum. Apparently you have not yet been introduced to summation notation. Since you are to use right-rectangles, you evaluate f(x) at the right side of each interval. The intervals need not be of equal widths, so there is no need for trying to take values at half-way points. So, the sum is

1 3 4 6 7 9 10
4 8 6 10 10 12 16

f(3)*(3-1) + f(4)*(4-3) + f(6)*(6-4) + f(7)*(7-6) + f(9)*(9-7) + f(10)*(10-9)
= 8*2 + 6*1 + 10*2 + 10*1 + 12*2 + 16*1
= 92

Well, Steve is basically saying that to estimate the value of the integral using 6 right rectangles, you need to add up the areas of those rectangles. Each rectangle will have a width of 1 unit because they are 1 unit apart. The heights of the rectangles will be determined by the values of f(x) at the corresponding x-coordinates.

So, for the first rectangle, the x-coordinate is 3 and the height can be estimated as f(3) = 8. The area of this rectangle is 1 (width) multiplied by 8 (height).

For the second rectangle, the x-coordinate is 4 and the height can be estimated as f(4) = 6. The area of this rectangle is 1 (width) multiplied by 6 (height).

You can continue to calculate the areas of the remaining rectangles using the same method. Once you have the areas, you can add them up to estimate the value of the integral.

Steve is explaining how to estimate the value of the integral using the right rectangle method. The formula for the area using the right rectangle method is:

Area = ∑ f(xk) ∆xk

In this formula, f(xk) represents the value of the function at each x-coordinate, and ∆xk represents the width of each rectangle.

To estimate the integral from 1 to 10, you need to add up the areas of each rectangle. In this case, you need to use 6 rectangles, so you will have 6 terms in the sum.

The x-coordinates given in the table are 1, 3, 4, 6, 7, 9, and 10, which correspond to the starting points of each rectangle. The widths of the rectangles are 2, 1, 2, 1, 2, and 1, as each rectangle has the same width.

To estimate the value of the integral, you need to plug in the values of f(x) from the table into the formula, along with the corresponding widths of each rectangle. Then, you add up all the terms in the sum.

So, in this case, you can calculate the value of the integral from 1 to 10 using the right rectangle method by adding up the following terms:

f(3) * 2 + f(4) * 1 + f(6) * 2 + f(7) * 1 + f(9) * 2 + f(10) * 1

Note that you are not required to know the values of f(2.5), f(5.5), or f(8.5) in order to estimate the integral using the right rectangle method.

Steve is explaining the process of estimating the value of the integral using right rectangles. The formula he mentioned,

6
∑ f(xk) ∆xk
k=1

is the finite sum approximation method for estimating the integral. Let me break it down for you:

1. Identify the given values of f(x) and their corresponding x-values. In this case, the values of x are 1, 3, 4, 6, 7, 9, and 10. The corresponding values of f(x) are 4, 8, 6, 10, 10, 12, and 16.

2. Determine the width of each rectangle (∆x) by subtracting the x-values of each pair of adjacent points. The ∆x values in this case are 2, 1, 2, 1, 2, and 1.

3. Multiply each f(x) value with its corresponding ∆x value to calculate the area of each rectangle.

4. Sum up the areas of all the rectangles using the formula mentioned by Steve.

In this case, since you only have the given values of f(x) at certain x-values, you may have to estimate the value of f(2.5), f(5.5), and f(8.5) using any appropriate method, such as linear interpolation or curve fitting.

Although it is possible to estimate the integral without those values by assuming certain patterns or approximations, it may not provide an accurate estimation. So it is recommended to estimate the missing values to obtain more reliable results.

81