Using rectangles whose height is given by the value of the...

Using rectangles whose height is given by the value of the function at the midpoint of the rectangle's base the midpoint rule estimate the area under the graphs of the following functions, using first two and then four rectangles. $$ f(x)=x^{3} \text { between } x=0 \text { and } x=1 $$

Using rectangles whose height is given by the value of the function at the midpoint of the rectangle's base the midpoint rule estimate the area under the graphs of the following functions, using first two and then four rectangles.
$$
f(x)=x^{3} \text { between } x=0 \text { and } x=1
$$

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Key Concepts

Midpoint Rule

The midpoint rule is a numerical integration method that approximates the area under a curve by dividing the interval into subintervals and using the function value at each subinterval's midpoint as the height of a rectangle to estimate its area.

Riemann Sums

Riemann sums are a class of methods for approximating the definite integral of a function, where the area under the curve is approximated by a sum of areas of chosen shapes, such as rectangles, over a partitioned interval.

Partitioning the Interval

This concept involves dividing the overall interval of integration into smaller, equal or unequal subintervals, which is a fundamental process in numerical integration methods and ensures that the approximation captures the behavior of the function over each segment.

Sample Point Selection

The practice of selecting specific points within each subinterval, such as the midpoint, is critical in numerical integration methods like the midpoint rule because it determines the height of the rectangles used in the approximation.

Numerical Approximation Techniques

These techniques provide methods for estimating the value of definite integrals when analytical integration is difficult, by summing estimated contributions from small intervals computed using formulas like the midpoint rule.

Transcript

00:01 This problem serves as a really great precursor to integration.

00:06 In this case, we are getting at the heart of the definition of what integral is.

00:11 And you might have heard the word rhyme in sum, or you might learn it soon after this.

00:17 But that is what we're doing.

00:19 Essentially, what we're doing is finding the area under a curve by splitting up that area into a different number of triangles, then adding the area of those triangles together to get the area under the area under the the curve and you'll be doing this with integrals and that's what an integral does is it finds that area under the curve.

00:38 So in this case we have the function f of x equals x cubed on our closed interval 0 .1.

00:46 So for part a we're told well let's find the area or really estimate the area using two rectangles.

00:54 So we have to find delta x.

00:56 So that's basically the length or the width of our rectangles under this curve.

01:02 And the way we find that is b minus a over the number of rectangles, n.

01:08 So in this case, we'll have 1 minus 0 over 2, and we'll get 1 over 2.

01:14 And then we're going to say let x1 and x2 be our midpoints.

01:18 So that would be where our rectangle hits the curve at the top of our rectangle.

01:24 So our area would be equal to f of x1 times delta x plus f of x2, times delta x...

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