Find the absolute value of the resulting error if the value of the integral from 0 to 3 of x cubed, dx is estimated with 3 circumscribed rectangles of equal width.

For a better understanding of the solution provided here, please check the attached file which contains the required diagram pertaining to the question.

The three circumscribed rectangles are shown in the diagram in light green color.

The width of each rectangle is 1 unit.

The height of each rectangle is determined by the equation of the curve under consideration, which is: . Thus the height of each rectangle is the value of the y-coordinate of the rectangle which, as can be clearly seen from the diagram, is 1 for the first rectangle, 8 for the second and 27 for the third rectangle. These values are derived by plugging in x=1,2,3 in the equation and finding out the corresponding value of y.

As can be clearly seen from the diagram, the area estimated by the circumscribed rectangles is an overestimation and thus will present an inherent error. Let us now proceed to calculate that error in absolute value as required by the question.

We know that the total area of the rectangles is the sum of the areas of each circumscribed rectangle. Thus:

Area of Rectangles=Area of 1st rectangle+Area of 2nd rectangle+Area of 3rd rectangle=

From integration we know that the actual area will be:

Actual Area =

Thus, the absolute value of the resulting error is:

Absolute Error=|Area of Rectangles - Actual Area|=

Thus, the Absolute Error is 15.75

d

step-by-step explanation:

hello and welcome!

so, when we see the words "proportional" along with a table, we need to know that we're looking numbers that divide by the same thing.

looking at this table, we simply have to do a trial and error, and divide to see which numbers have the same quotient. if you look at choice a & c, they both have equal quotients for three of the numbers, with one exception, so those can't be the answer. looking at b, there is no similarity, so that can't be it. then, we see d; if you divide all the numbers in the table, you'll see that they all divide by two, which is a proportional relationship, and our answer!

also, always remember that the x will always be the number that goes into y, and somewhere behind this table, is a formula. for this specific table, the formula would be:

since our x-value is being divided by two to get the y-value. and if they were to ask for more numbers in the series, this is the formula you would use to find them.

step-by-step explanation:

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