The table below gives selected values for the function f(x). With 5 rectangles, using the left side of each rectangle to evaluate the height of each rectangle, estimate the value of the integral from 1 to 2 of f(x)dx.

x 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0
f(x) 1 0.909 0.833 0.769 0.714 0.667 0.625 0.588 0.556 0.526 0.500

a) 0.7456 <----My answer, can you check if this is correct pls? Thanks
b) 0.6456
c) 0.6919
d) 0.6932

left hand [0.2] [ 1 + 0.833 + 0.714 + 0.625 + 0.556 ]

[0.2] [3.72] = 0.74

okay but I am right with answer A, right?

Again, because its difficult to understand the numbers on the top

x = 1, 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9, 2.0
f(x)=1, 0.909, 0.833, 0.769, 0.714, 0.667, 0.625, 0.588, 0.556, 0.526, 0.500

I get .7187

add row 2 from 1 to .526 and multiply by delta x = .1

well I got closer to .72

look, from 1 to 1.1 you use the left value of y which is 1

from 1.1 to 1.2 you use the left value of y which is .909
from 1.2 to 1.3 you use the left value of y which is .833
etc
there are ten of them and 10 * delta x = 1.0000

you never use the point ( 2.00, 0.500) because it is on the far right.

Draw it !

Oh, sorry, did not see use just 5 rectangles, you are right

To estimate the value of the integral from 1 to 2 of f(x)dx using 5 rectangles and the left side of each rectangle, you can follow these steps:

1. Calculate the width of each rectangle by dividing the total width (2 - 1 = 1) by the number of rectangles (5). The width of each rectangle is 1/5 = 0.2.

2. Find the height of each rectangle using the left side of each interval in the table. For the first rectangle, the left side is 1, so the height will be f(1) = 1. For the second rectangle, the left side is 1.2, so the height will be f(1.2) = 0.833. Continue this process for each rectangle.

3. Multiply the width of each rectangle by its corresponding height to find the area of each rectangle.

4. Add up the areas of all the rectangles to estimate the value of the integral.

Let's go through the calculations:

Width of each rectangle (Δx) = 1/5 = 0.2

Height of each rectangle:
Rectangle 1: f(1) = 1
Rectangle 2: f(1.2) = 0.833
Rectangle 3: f(1.4) = 0.714
Rectangle 4: f(1.6) = 0.625
Rectangle 5: f(1.8) = 0.556

Areas of each rectangle:
Rectangle 1: (0.2) * (1) = 0.2
Rectangle 2: (0.2) * (0.833) = 0.1666
Rectangle 3: (0.2) * (0.714) = 0.1428
Rectangle 4: (0.2) * (0.625) = 0.125
Rectangle 5: (0.2) * (0.556) = 0.1112

Adding up the areas of all the rectangles: 0.2 + 0.1666 + 0.1428 + 0.125 + 0.1112 = 0.7456

Therefore, the estimated value of the integral from 1 to 2 of f(x)dx, using 5 rectangles and the left side of each rectangle, is 0.7456.

So your answer, 0.7456, is correct (Option a). Well done!