Definite Integral Questions

Question

I have a few questions that deal with definite integrals that I am confused about.

 

1) (a) Find the Riemann sum for f(x)=x^3, 2≤x≤12, if the partition points are 2,5,9,12 and the sample points are 3,8,10.

I found the answer for this part to be 5641.

I am stuck on part b:

(b) Find the Riemann sum if the partition points are 2,5,9,12 and the sample points are the midpoints.

 

 

2) A table of values of a decreasing function ff is shown. Use the table to find a lower and an upper estimate for the definite integral from 8 to -10 of f(x) dx. There is a chart & I will do my best to relay it. 

 

x=     -10   -7   -4   -1   2    5    8

f(x)=    9    5    2    -3   -4  -8   -15

 

Lower Estimate?

Upper Estimate?

 

 

3) The following sum

 

sin(10+(5/n)⋅(5/n) + sin(10+(5/n)⋅(5/n) + ... + sin(10+(5/n)⋅(5/n) is a right Riemann sum for the definite integral of b to 10 (b being the upper number) f(x) dx.

 

I found b to be 15 & f(x) to be sin(x).

 

The second part of the problem asks...

It is also a Riemann sum for the definite integral c to 0 (c being the top number) g(x) dx.

 

Where c equals? 

I found this to be 5.

and g(x) equals?

Update: As I was working through this one again I found the correct answer to be sin(10 + x) but I'm uncertain as to why. Could someone give me an explanation of this?

 

 

4) Given that 7 ≤ f(x) ≤ 8 for -5 ≤ x ≤ 5, estimate the value of ∫ from 5 to -5 of f(x) dx. 

 

The answer has two blanks to put the answer in...

__ ≤ ∫ from 5 to -5 of f(x) dx ≤ __

 

Then the last question I have seems to be practically the same concept...

 

Use property 8 to estimate the value of ∫ from 11 to 2 of 6/x dx. 

 

__ ≤ ∫ from 11 to 2 of 6/x dx ≤ __

Answer 1

4) Given that 7 ≤ f(x) ≤ 8 for -5 ≤ x ≤ 5, estimate the value of ∫ from 5 to -5 of f(x) dx. This just means that the y values of the function are all bedtween 7 and 8, over the interval [-5,5]. If we use the lowest possible y, then the def. integral is the area of a rectangle having base length 10, height 7, or minimal area under curve is 70.  Using max. possible y of 8, then the area maximum would be 10(8).

The answer has two blanks to put the answer in...
70 ≤ ∫ from 5 to -5 of f(x) dx ≤ 80.


2) A table of values of a decreasing function f is shown. Use the table to find a lower and an upper estimate for the definite integral from 8 to -10 of f(x) dx. There is a chart & I will do my best to relay it.

x= -10 -7 -4 -1 2 5 8
f(x)= 9 5 2 -3 -4 -8 -15

Lower Estimate?
Upper Estimate?

Using similar reasoning that I employed 4) above, I will do the estimates for from -10 to 8, i.e. normal L to R order of areas of rectangles as an area estimate. Note that the rectangles will all have Δx = 3. In this first estimate, I'll use left endpoints:

First Est. = 9(3) + 5(3) + 2(3) + -3(3) + -4(3) + -8(3) ...(you do the arithmetic!)

 

Second Est. using right endpoints: =  5(3) + 2(3) + -3(3) + -4(3) + -8(3) + -15(3) ...

 

Now, a word of caution: The question wanted the def. integration to proceed from R to L, so the above answers must have their signs changed! The lower final estimate would be the most negative of these revised two numbers.