Use the Midpoint Rule with n=6 to estimate the value of the integral. ∫0^3 √(x^2+1) d x

Name: Problem 2, Chapter 4: Calculus: Early Transcendental Functions
Uploaded: 2022-10-06T01:47:36-08:00
Duration: 6 min 29 s
Channel: Gabrielle Lee
Description: Use the Midpoint Rule with n=6 to estimate the value of the integral. ∫0^3 √(x^2+1) d x

Use the Midpoint Rule with n=6 to estimate the value of the...

Key Concepts

Definite Integral Approximation

Definite integral approximation is the process of estimating the value of an integral over a specified interval, often using methods such as the Midpoint Rule, trapezoidal rule, or Simpson's rule. This concept is important in many practical applications where an analytic solution is difficult or impossible to derive.

Partitioning the Interval

Partitioning the interval involves dividing the interval of integration into a number of equal subintervals, which is essential for applying numerical integration methods like the Midpoint Rule. The equal-width subintervals facilitate a systematic approach to approximating the integral.

Numerical Integration

Numerical integration involves approximating the value of a definite integral using discrete sums instead of finding an antiderivative analytically. This approach is useful for complex functions or when an exact integral is difficult to obtain.

Midpoint Rule

The Midpoint Rule is a numerical technique for estimating the area under a curve by dividing the interval of integration into equal subintervals and using the function's value at the midpoint of each subinterval to approximate the area. This method often provides a good balance between simplicity and accuracy.

Transcript

00:01 Okay, so for this case, we need to estimate the value of the integral using the midpoint rule, where n equals to 6.

00:14 So the midpoint rule assumes that if f of x is continuous from a to b, then the integral form of f of x from a to b is equal to the limit of mn, where n approaches infinity.

00:47 And our mn here is given by the formula n i equals to 1fmi delta x, where delta x is b minus a divided by n and mi is the midpoint of the n -sub -interval, where a and b is divided into n -sub -intervals with the length of delta -x.

01:21 So it'll be easier to understand what this mid -point rule is once we start working on a question.

01:30 But what we can do here is we can actually simplify this equation.

01:33 So delta -x is actually going to be a constant number.

01:36 So what you can do is you can take the delta x out of the series.

01:42 So it would simply be delta x of n.

01:49 Then in the end, you'll just have to add up f at the different intervals or the midpoint subinterval.

02:00 So our question is to estimate the value of the integral x square root of x squared plus 1 from 0 to 3.

02:15 So once again we are going to determine delta x which is given as b minus a over n so it be 3 minus 0 over 6 because we are asked to solve it with the midpoint rule where n equals to 6 and our b here is the upper limit and a is the lower limit.

02:46 So this simplifies to 1 over 2.

02:50 So our limit goes around 03, and the sub -intervals need to have an interval of 1 over 2.

02:57 So it will start up 0, and it's going to increase by 1 over 2.

03:04 And then from 1 over 2, it's going to become 1.

03:07 It's going to become 1 because it increases by 1 over 2.

03:11 And then three over two and then full oh whoops two and then two and then two and five over two and three so now that gives us a total of six sub intervals like the question asked now our next step is that we need to find the midpoint um the midpoint of the sub interval okay so so the midpoint from 0 to 1 over 2 is going to be 1 over 4...

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