How do you find an approximation to the integral #int(x^2-x)dx# from 0 to 2 using a Riemann sum with 4 subintervals, using right endpoints as sample points?

Answer 1

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

#int_0^2 (x^2-x)dx#
Note that #f(x) = x^2-x# and #a=0# and #b=2#
#n=4#
So #Deltax = (b-a)/n = (2-0)/4 =1/2#
All endpoints: start with #a# and add #Deltax# successively:
#0# #underbrace(color(white)"XX")_(+1/2)# #1/2# #underbrace(color(white)"XX")_(+1/2)# #1# #underbrace(color(white)"XX")_(+1/2)# #3/2# #underbrace(color(white)"XX")_(+1/2)# #2#
Right endpoints: #1/2#, #1#, #3/1#, #2#
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(1/2)*1/2+f(1)*1/2+f(3/2)*1/2+f(2)*1/2#
#= (f(1/2)+f(1)+f(3/2)+f(2))1/2#
#R = ((-1/4)+(0)+(3/4)+(2))1/2 = 5/2*1/2 = 5/4#
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Answer 2

The approximation to the integral ( \int_{0}^{2} (x^2 - x) , dx ) using a Riemann sum with 4 subintervals and right endpoints as sample points is ( \frac{1}{4}[(1)^2 - (1)] \cdot \frac{2}{4} + [(1.5)^2 - (1.5)] \cdot \frac{2}{4} + [(2)^2 - (2)] \cdot \frac{2}{4} ). This simplifies to ( \frac{1}{4} + \frac{9}{8} + 1 ), which equals ( \frac{15}{4} ).

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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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