the intergral
(10
4 (2x^2+3x+1)dx
i have to find the sum of the intergral using right and left endpts when n=3. I am confused on how to even start this or how to use these endpts.
What do you mean by the sum of the integral? Do you mean the value of the integral? Are the limits of integration 4 to 10? What is n? It does not appear in the mathematial expression you have written. Are you supposed to be doing a Simpson's Rule numerical integration with n=3 terms?
That is what I assume they want you to do.
The exact value of the integral is
(2/3)x^3 + (3/2)x^2 +x evaluated at x=10, minus the value at x = 4. This is
826.67 - 70.67 = 756
Using Simpson's rule with 3 terms and h=3 as the interval width, I get
(h/3)[f(4) + 4 f(7) + f(10)]
= 1*[45 + 480 + 231) = 756
The agreement is perfect because the curve is a parabola, and Simposn's rule provides a parabolic best fit to each interval.
Find the Riemann sum for this integral using right and left endpoints and n=3. when i type in 756 it shows to be the wrong answer.
You left out the word Riemann before; so I solved the wrong problem.
The Riemann sum is defined here:
http://en.wikipedia.org/wiki/Riemann_integral
It is a less accurate approxiation to the integral than Simpson's rule. I will leave you to calculate it.
To find the sum of the integral using right and left endpoints, you need to use the Riemann sum approximation method. This method divides the interval of integration into smaller subintervals and approximates the area under the curve using rectangles.
To start, you need to determine the interval of integration. In this case, the interval is from 4 to 10.
Next, calculate the width of each subinterval. Since n = 3, there will be three subintervals. The width of each subinterval is given by:
Δx = (b - a) / n = (10 - 4) / 3 = 2
So, each subinterval will have a width of 2.
Now, you need to determine the x-values of the right and left endpoints of each subinterval. Since the width is 2, the x-values for the right endpoints will be 6, 8, and 10. The x-values for the left endpoints will be 4, 6, and 8.
Finally, you can calculate the area of each rectangle by evaluating the function at each x-value and multiplying it by the width. For example, for the right endpoint of the first subinterval:
f(6) = 2(6)^2 + 3(6) + 1 = 73
Area of the rectangle = f(6) * Δx = 73 * 2 = 146
Repeat this process for the remaining x-values and add up all the areas of the rectangles to get the sum of the Riemann sum approximation.
Note that as n approaches infinity, the Riemann sum approximation approaches the exact value of the integral. However, with a small value of n, the Riemann sum will only provide an approximation of the integral.