Numerical integration

Think of it like trying to measure the area of a puddle that has a really wiggly edge. You can't just use a ruler and multiply length by width. Integration (which you've learned about) is the mathematical way to find the exact area under a curve. But sometimes, the curve is so messy, or we only have a few points of data, that finding the exact area is impossible or super difficult.

That's where numerical integration comes in! It's a fancy way of saying we're going to estimate that area using simpler shapes. Instead of trying to find the exact area of the whole wiggly puddle at once, we chop it up into many tiny, easy-to-measure pieces, like rectangles or trapezoids. Then, we find the area of each little piece and add them all up. The more pieces we use, the closer our estimate gets to the real area!

• The Big Idea: We're replacing a complicated curve with simpler shapes (like straight lines) to estimate the area underneath it.
• Why? Because sometimes the exact answer is too hard to find, or we only have data points, not a formula.
• Goal: Get a really good approximation of the area.

Imagine you're driving your car, and you want to know how far you've traveled. Your speedometer tells you your speed at any given moment, but your speed is constantly changing – you speed up, slow down, stop at lights, etc. If you had a graph of your speed over time, the area under that speed graph would tell you the total distance you traveled.

Now, imagine your speedometer is broken, and you can only read it every 5 minutes. You have a list of speeds at specific times, but no smooth curve. How do you find the total distance?

1. Collect Data: You write down your speed every 5 minutes: 0 mph, 30 mph, 40 mph, 20 mph, 0 mph.
2. Draw It Out: If you plot these points on a graph, it looks like a bumpy line.
3. Estimate with Shapes: You can't use an exact formula. So, you draw rectangles or trapezoids under each segment of your speed graph. For example, for the first 5 minutes, you might assume you were going 0 mph, then suddenly jumped to 30 mph. A rectangle or trapezoid helps estimate the distance during that 5-minute chunk.
4. Add Them Up: You calculate the area of each small rectangle or trapezoid (which represents the distance traveled in that small time interval) and add them all together. This sum gives you a good estimate of your total travel distance. This is exactly what numerical integration does!

Let's break down how we use those simple shapes to estimate the area under a curve. We'll focus on two main methods: Riemann Sums (using rectangles) and the Trapezoidal Rule (using trapezoids).

1. Divide the Interval: First, you take the total width of the area you want to measure (called the interval) and slice it into many smaller, equally sized pieces. Think of it like cutting a long loaf of bread into many slices.
2. Choose Your Shape: Decide if you're going to use rectangles (Riemann Sums) or trapezoids (Trapezoidal Rule) to approximate the area of each slice.
3. Riemann Sums (Rectangles): For each slice, you draw a rectangle. The width of the rectangle is the width of your slice. The height of the rectangle is determined by the curve at a specific point within that slice (left endpoint, right endpoint, or midpoint).
4. Trapezoidal Rule (Trapezoids): For each slice, you draw a trapezoid. The parallel sides of the trapezoid are the heights of the curve at the left and right edges of your slice. The width of the trapezoid is the width of your slice.
5. Calculate Area of Each Shape: Find the area of each individual rectangle or trapezoid. Remember, the area of a rectangle is width × height, and the area of a trapezoid is (1/2) × (height1 + height2) × width.
6. Sum Them Up: Add all those tiny areas together. This sum is your numerical approximation of the total area under the curve. The more slices you make, the better your estimate usually is!