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

Using rectangles each of 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. $$ f(x)=4-x^{2} \text { between } x=-2 \text { and } x=2 $$

Using rectangles each of 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.
$$
f(x)=4-x^{2} \text { between } x=-2 \text { and } x=2
$$

Key Concepts

Subinterval Partitioning

Subinterval partitioning refers to dividing the overall integration domain into smaller, equally or unequally spaced subintervals. This division allows for a more manageable and often more accurate approximation of the area under a curve when using methods like Riemann sums or the midpoint rule. Increasing the number of subintervals generally improves the approximation accuracy.

Numerical Integration

Numerical integration is a collection of algorithms used to approximate the value of a definite integral when an analytical solution is challenging or impossible to obtain. Techniques such as the midpoint rule, trapezoidal rule, and Simpson's rule are practical applications of numerical integration, enabling the estimation of areas, accumulated totals, and other integral-related quantities in various scientific and engineering problems.

Midpoint Rule

The midpoint rule is a method for estimating the value of a definite integral by dividing the integration interval into equal subintervals, taking the midpoint of each subinterval, evaluating the function at these midpoints, and multiplying by the width of each subinterval. This rule often provides a better estimate than using the endpoints because it can better capture the behavior of the function over each interval.

Riemann Sums

Riemann sums are a foundational concept in calculus used for approximating the area under a curve (i.e., the definite integral) by summing the areas of shapes such as rectangles. The method involves partitioning the interval into smaller segments, choosing a sample point within each segment, and multiplying the function value at that point by the segment's width.

Transcript

00:01 Question is asking us to estimate the area beneath this curve using the midpoint rule.

00:06 So for all these problems again, we want to find our delta x first.

00:11 So for that, i want to subtract my two endpoints from each other, divided by the number of rectangles i want.

00:19 For this first part, we just want two.

00:22 And when you do that, you should find that delta x is equal to 2.

00:28 So i want to go to my graph.

00:29 I'm going to start negative 2.

00:30 If i go over two, i'm just at zero, go over two again, and back at my two.

00:36 So for the midpoint rule, which is a little bit different, our height of our rectangles that we're going to draw is actually going to come from the midpoint between these different intervals that we just created.

00:50 So the middle point between zero and two is just one, and here it's negative one.

00:58 So if i draw my rectangles, this is going to be my height there.

01:03 And then i'm still going to go my full length that i found.

01:07 So two.

01:10 Then same thing on the other side.

01:14 So then that's going to become my rectangle.

01:18 And i again, i'm just going to add up the areas of both of those rectangles.

01:22 So for that, i'm going to have my length times.

01:29 Now for this, i'm putting in my height.

01:31 So that's just that midpoint that i just found.

01:33 The 1 and negative 1.

01:34 I'm going to plug that into my function...