Use the trapezium rule with four equal intervals to find the approximate values of 𝐚 ∫0^1 e^-x^2

Use the trapezium rule with four equal intervals to find the...

Key Concepts

Trapezoidal Rule

The trapezoidal rule is a numerical integration technique that approximates the area under a curve by dividing the interval into smaller segments and estimating the area of each segment using a trapezoid. The method uses the values of the function at the endpoints of these subintervals to form a linear approximation, calculating the area as the average of the endpoints times the subinterval width.

Equal Interval Partitioning

This concept involves dividing the integration interval into subintervals of equal width, which simplifies the application of numerical methods like the trapezoidal rule. With equal intervals, the step size remains constant, making it easier to apply the formula consistently across the entire interval.

Definite Integral

A definite integral represents the accumulation of quantities such as area under a curve between specified limits. It is a fundamental concept in calculus, defining the total accumulation of a function's values over an interval.

Numerical Integration

Numerical integration encompasses techniques designed to approximate the value of definite integrals when an analytical solution is challenging or impossible to obtain. Methods like the trapezoidal rule offer practical means to compute integrals, balancing complexity with accuracy.

Transcript

00:01 We are going to use the trapezoidal rule, the midpoint rule and simpson's rule to approximate the interval from 0 to pi half of cubic root of 1 plus cosine of x with force of intervals.

00:18 So let's define the function f of x equal cubic root of 1 plus cosine of x on the interval 0 by half.

00:44 And the lower limit of integration is 0, a equals 0.

00:50 The upper limit is b equal pi half.

00:53 The numbers of intervals n is 4.

00:57 And with that, we can calculate the step size delta x equal b minus a over n, which is equal to pi half minus 0 over 4, that is pi over 8.

01:14 So that's the step size, pi over 8.

01:26 With that, we can now write the expression for the nodes.

01:31 We'll be using xy, x, sorry, equals a plus i times delta x, that is 0 plus i times pi over 8, and that is i pi over 8.

01:49 4 i equals 0 1 up to 4 that is in this case we can write all the values now not too much of them 0 1 2 3 and 4 as the 5 value that takes i and the midpoints we will be using fault trap so and the midpoint rule we get x x x x bar is x sub i plus x sub i plus 1 over 2 that is is the average of the two consecutive nodes and that's equal to using this expression we found before is i by over 8 plus i plus 1 by over 8 and that over 2 and if we simplify this, we get 2i plus 1 pi over 16.

03:10 4, in this case i from 0 up to 3, because we have force midpoints because we have force of intervals.

03:27 0, 1, 2, 3...

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