How do you find an approximation for the definite integrals #int 1/x# by calculating the Riemann sum with 4 subdivisions using the right endpoints from 1 to 4?

Answer 1

Assuming equal subdivisions, see the explanation section below.

I will use what I think is the usual notation throughout this solution.

#int_1^4 1/x dx#
Note that #f(x) = 1/x# and #a=1# and #b=4#
#n=4# So #Deltax = (b-a)/n = (4-1)/4 =3/4#
To find all of the endpoints: start with #a# and add #Deltax# successively:
#1# #underbrace(color(white)"XX")_(+3/4)# #7/4# #underbrace(color(white)"XX")_(+3/4)# #10/4# #underbrace(color(white)"XX")_(+3/4)# #13/4# #underbrace(color(white)"XX")_(+3/4)# #16/4#
Right endpoints: #7/4#, #5/2#, #13/4#, #4#
Now the Riemann sum is the sum of the area of the 4 rectangles. We find the area of each rectangle by #"height" xx "base" = f("endpoint") xx Deltax#

So

#R = f(7/4)*1+f(5/2)*1+f(13/4)*1+f(4)*1#
#= 4/7+2/5+4/13+1/4#

To finish, do the arithmetic.

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Answer 2

To approximate the definite integral ∫(1 to 4) 1/x by calculating the Riemann sum with 4 subdivisions using the right endpoints, you first divide the interval [1, 4] into 4 equal subintervals. The width of each subinterval is Δx = (4 - 1) / 4 = 3/4.

Then, you evaluate the function 1/x at the right endpoint of each subinterval and multiply by the width of the subinterval. This gives you the height of a rectangle for each subinterval, and multiplying by the width gives you the area of each rectangle.

Finally, you sum up the areas of all the rectangles to find the approximate value of the integral.

The right endpoints of the intervals are: 1 + Δx, 1 + 2Δx, 1 + 3Δx, and 1 + 4Δx.

So, you evaluate the function 1/x at these points:

1 + Δx = 1 + (3/4) = 7/4 1 + 2Δx = 1 + 2(3/4) = 5/2 1 + 3Δx = 1 + 3(3/4) = 11/4 1 + 4Δx = 1 + 4(3/4) = 4

Then, you calculate the area of each rectangle:

Area of 1st rectangle: (3/4) * (1/(7/4)) Area of 2nd rectangle: (3/4) * (1/(5/2)) Area of 3rd rectangle: (3/4) * (1/(11/4)) Area of 4th rectangle: (3/4) * (1/4)

Finally, you sum up the areas of all the rectangles to find the approximation for the integral.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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