Use the definition of the definite integral (with right hand rule) to evaluate

Using these formulas, you can simplify the expression to:20∑ i2/n2 - 45∑ i/n + 10And then plug in the limits of integration:20[(22^2 - (-3)2)/n2] - 45[(2 + (-3))(n+1)/2n] + 10= 20[(4 + 9)/n2] - 45[-n/2n] + 10= 20(13/n2) + 45/2 + 10= 45/2 + 260/n2In summary, to evaluate the definite integral ∫(4x2-9x+2)dx from -3
  • #1
macilath
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0

Homework Statement


Use the definition of the definite integral (with right hand rule) to evaluate the following integral from -3 to 2
[tex]\int(4x^2-9x+2)dx[/tex]

Homework Equations


[tex]\int[/tex] from a to b of f(x)dx = limit as [tex]n\rightarrow[/tex][tex]\infty[/tex] of [tex]\sum f(xi)\Deltax[/tex]. i = 1

The Attempt at a Solution


I found delta x = (b-a)/n, so delta x = 5/n.
Then,
limit as [tex]n\rightarrow[/tex][tex]\infty[/tex] of [tex]\sum (4(i/n)^2-9(i/n)+2)(5/n)[/tex].
I distributed the (5/n) out, and a little algebra later, got that
limit as [tex]n\rightarrow[/tex][tex]\infty[/tex] of [tex]\sum ((20i^2)/n^3)-(45i/n^2)+(10/n)[/tex].
This is where I get stuck, I'm not sure how to simplify this to evaluate the limit.

Thanks for any help!

Edit: Sorry for sloppy forum code. LaTEX is new to me.
 
Last edited:
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  • #2
Assuming you've done all the math right up to this point, you just need some summation formulas to finish evaluating the integral.

k = nk

i = n(n+1)/2

i2 = n(n+1)(2n+1)/6
 

1. What is the definition of the definite integral?

The definite integral is a mathematical concept that represents the area under a curve on a graph. It is used to find the total accumulation of a function over a given interval.

2. What is the right hand rule in the context of definite integrals?

The right hand rule is a method for approximating the definite integral by dividing the area under the curve into rectangles and using the right endpoint of each rectangle to determine its height.

3. How do you evaluate a definite integral using the right hand rule?

To evaluate a definite integral using the right hand rule, you first divide the interval into equal subintervals. Then, you use the right endpoint of each subinterval to determine the height of the rectangle. Finally, you add up all the areas of the rectangles to get an approximation of the definite integral.

4. What are the limitations of using the right hand rule to evaluate a definite integral?

The right hand rule can only provide an approximation of the definite integral and is not always accurate. It also assumes that the function being integrated is continuous and does not take into account any irregularities in the shape of the curve.

5. How can you improve the accuracy of the right hand rule when evaluating definite integrals?

One way to improve the accuracy is by increasing the number of subintervals used in the approximation. This will result in smaller rectangles and a closer approximation to the actual value of the definite integral. Another method is to use other techniques such as the trapezoidal rule or Simpson's rule.

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