Round off your calculations to four decimal places. Estimate...

Key Concepts

Numerical Integration

Numerical integration is a collection of algorithms for approximating the value of definite integrals. It is often used when an analytical solution is difficult or impossible to obtain. The strategy involves evaluating the integrand at a set of discrete points and combining these evaluations to estimate the overall area under the curve.

Midpoint Rule

The midpoint rule is a numerical method that approximates the area under a curve by using the value of the function at the midpoint of each subinterval. This method is simple and provides improved accuracy over the basic rectangular method, especially when the function behaves well over each subinterval.

Trapezoidal Rule

The trapezoidal rule approximates an integral by replacing the region under the curve with trapezoids rather than rectangles. This method uses the average of the function's values at the endpoints of each subinterval, leading to a linear approximation that often produces better results than the midpoint or rectangular methods, particularly for functions that are approximately linear over small intervals.

Simpson's Rule

Simpson's rule is a method for numerical integration that approximates the integrand by a second-degree polynomial on each pair of subintervals. This method leverages parabolic fitting over the intervals, usually resulting in a higher degree of accuracy compared to both the midpoint and trapezoidal rules, especially when the integrand is smooth.

Subinterval Partitioning

Subinterval partitioning involves dividing the overall interval of integration into smaller, equal parts. The number of subintervals, often denoted by n, impacts the accuracy of numerical integration techniques. A larger number of subintervals generally leads to a more accurate approximation, as it allows the numerical method to better capture the behavior of the function over the interval.

Rounding and Accuracy in Numerical Methods

Rounding and accuracy in numerical methods refer to the practice of adjusting computed results to a specified number of decimal places. This is crucial in numerical methods to ensure consistency and clarity in results, particularly when approximating integrals where inherent approximation error exists due to discretization and finite computation precision.

Transcript

00:01 We will be estimating the integral from 1 to 2 of e to the x over x, that is the exponential of x divided by x.

00:13 First by the midpoint estimate with 4 sub -intervals in part a, then after that trapezoid rule with 8 sub -intervals in part b, and finally in part c, simpson's rule with 4 sub -intervals.

00:29 We will round off the results of 4 decimal places.

00:33 So let's start by defining the function we are integrating e to the x over x, let's call it f of x, and we consider that function defined on the interval of integration, the close interval from 1 to 2.

01:01 So in part a we're going to use midpoint estimate with 4 sub -intervals.

01:06 We remember the general expression for a midpoint mn, that is for n sub -intables, is p minus a over n times the sum of the images of the midpoint the first one is x0 plus x1 over 2 and up to the last midpoint the image of the last midpoint is x n minus 1 plus f at sorry i mean x n minus 1 plus x n over 2 so is b minus a over n times the sum of the images of the midpoints of this sub -interables.

01:54 So here using four sub -intables n -equal four, we can calculate the step -size age, which is a common distance between two consecutive nodes, and is calculated that as b -minus a over n, it's just this factor here, and in this case is 2 minus 1 over 4, that is 1 fourth.

02:22 So h, the step size, is one fourth.

02:26 And with that, we can write a general expression for the notes.

02:30 Let's say xi equal the left end point of the interval of integration, 1 plus i times h, and that is 1 plus i over 4, which we can write as i plus 4 over 4.

02:50 And the index i goes from 0 up to 4.

02:57 Now the midpoints of the subinterables determined by the nodes, let's say they are xi bar, is the average of the nodes xi and xi plus 1.

03:18 And we use the formula here for the nodes x i plus 4 over 4.

03:25 Plus x i plus 1 will be i plus 1 plus 4 that is i plus 5 divided by 4 and all that divided by 2 we finally get 2 i plus 9 that divided by 8 so this in this case the index i goes from 0 up to 3 so with that we can now say that the midpoint estimate with 4 sub intervals is equal to b minus a over n as we saw here is h so is h times f at the first midpoint is x0 bar plus f at the second midpoint is one x1 bar plus f at x2 bar plus f at x3 bar so m4 is h is one -four times f at the first midpoint x0 bar is obtained by putting i equals 0 here we get 9 over 8 plus f at x1 bar is putting i equal 1 here we get 11 over 8 plus x2 bar is putting i equal 2 we get 13 over 8 plus the image at x3 bar is putting i equal three we get 15 over 8 now we can apply the function so m4 is one -fourth times so it's the exponential of 9 over 8 divided by 9 over 8 so we get 8 exponential of 9 over 8 by 9 plus the exponential of 11 over 8 divided by 8 finally is 8 e to the 11 8th divided by 11.

06:24 Here similarly we get 8 e to the 13 8th divided by 13 plus 8 e to the 15 8th divided by 15.

06:44 And then we can say that the midpoint estimate before so interval is equal 2.

06:52 We see an 8 common factor in all these terms that can be get out of the integral and simplify with the 4 in the denominator here and we get 2 times e to the 9 8th over 9 plus e to the 11 8th over 11 plus e to the 13 8 over 13 plus e to the 15 8th over 13 plus e to the 15 8 over 15.

07:32 Now here we got to use a calculator to find m4 and we found that m4 the midpoint estimate with force of intervals about 3 .0543.

07:51 So this is the approximation by using midpoint estimate with force of intervals.

07:58 And now we go to part b where we are going to use trapezoid the rule with 8 sub -interables.

08:05 So remember the general expression for chapter of the rule with n -sub -interples is b minus a over 2n times the image of the first node x0 plus 2 -1 plus 2 image of x1 plus up to to image of x -n minus 1 plus f at the last node x -n...

Please give Ace some feedback