A Method For Approximating the Areas of Irregular Regions Riemann Sums A Method For Approximating the Areas of Irregular Regions

The Necessity for Approximation In previous courses, you’ve learned how to find the areas of regular geometric shapes using various formulas…

The Necessity for Approximation However, when we have shapes like these, there are no nice, neat formulas with which to calculate their area…

The Necessity for Approximation To address this issue, we use a large number of small rectangles to approximate the area of one of these irregular regions. We may choose to use rectangles with different widths or rectangles with the same width.

The Necessity for Approximation Things to Remember When we know a function, it is best to approximate the area using rectangles with the same width. When we only know certain points of the function, we will let those points dictate the widths of the rectangles that we use.

The BIG Unanswered Question??? We’ve talked about the widths of the rectangles that we will use for approximation, but how will we decide what the heights of these rectangles will be?

Determining the Heights of our Rectangles Basically, unless specified, we can use any point on the function in the given interval for the height of a rectangle. However, we typically choose to use one of 3 points: The Left Endpoint The Right Endpoint The Midpoint

BEWARE OF SCARY NOTATION Caution! BEWARE OF SCARY NOTATION 11/28/2018

In Left Riemann sum, the left-side sample of the function is used as the height of the individual rectangle.

In Right Riemann sum, the right-side sample of the function is used as the height of the individual rectangle.

In the Midpoint Rule, the sample at the middle of the subinterval is used as the height of the individual rectangle.

In English Please!!! 11/28/2018

Left Hand Rectangular Approximation Method As its name indicates, in the Left Hand Rectangular Approximation Method (LRAM), we will use the value of the function at the left endpoint to determine the heights of the rectangles.

Left Hand Rectangular Approximation Method

Left Hand Rectangular Approximation Method

Right Hand Rectangular Approximation Method As its name indicates, in the Right Hand Rectangular Approximation Method (RRAM), we will use the value of the function at the right endpoint to determine the heights of the rectangles.

Right Hand Rectangular Approximation Method

Right Hand Rectangular Approximation Method

Midpoint Rectangular Approximation Method As its name indicates, in the Midpoint Rectangular Approximation Method (MRAM), we will use the value of the function at the midpoint to determine the heights of the rectangles.

Midpoint Rectangular Approximation Method

Midpoint Rectangular Approximation Method

Approximations from Numeric Data Time Speed 3 8 5 14 11 7 15 18 These methods for approximation are most useful when we don’t actually know a function, but where we have several numerical data points that have been collected. In this situation, the widths of our rectangles will be determined for us (and may not all be the same).

Approximations from Numeric Data To find the distance traveled in this situation, we can choose to use either LRAM or RRAM (but not MRAM since we don’t know the values at the midpoints. However, unlike our previous examples, the widths of our rectangles will all be different. Time Speed 3 8 5 14 11 7 15 18

Approximations from Numeric Data To find LRAM, we will use the function value at the left endpoint and the varying widths of rectangles. Time Speed 3 8 5 14 11 7 15 18

Approximations from Numeric Data Time Speed 3 8 5 14 11 7 15 18 To find RRAM, we will use the function value at the right endpoint and the varying widths of rectangles.

What if I HATE Rectangles?? Can I use another shape? More importantly, is there a shape that gives a better approximation? Let’s look at an example that explores a new approximation 11/28/2018

Actual area under curve:

Left-hand rectangular approximation: Approximate area: (too low)

Right-hand rectangular approximation: Approximate area: (too high)

Averaging the two: 1.25% error (too high)

Averaging right and left rectangles gives us trapezoids:

(still too high)

Trapezoidal Rule: ( h = width of subinterval ) This gives us a better approximation than either left or right rectangles.

Compare this with the Midpoint Rule: Approximate area: 0.625% error (too low) The midpoint rule gives a closer approximation than the trapezoidal rule, but in the opposite direction.

Ahhh! Oooh! Wow! Trapezoidal Rule: 1.25% error (too high) Midpoint Rule: (too low) 0.625% error Notice that the trapezoidal rule gives us an answer that has twice as much error as the midpoint rule, but in the opposite direction. Ahhh! If we use a weighted average: This is the exact answer! Oooh! Wow!

We got properties for indefinite integrals…. What now?? Do the same rules apply? What if the curve goes below the axis? Let’s do some examples!! 11/28/2018

Negative Values If f (x) is positive for some values of x on [a, b] and negative for others, then the definite integral symbol y = f (x) B a represents the cumulative sum of the signed areas between the graph of f (x) and the x axis, where areas above are positive and areas below negative. A b

Examples Calculate the definite integrals by referring to the figure with the indicated areas. Area A = 3.5 Area B = 12 B a A b c y = f (x)

Definite Integral Properties

Examples Assume we know that Then A) B)

Examples (continued) C) D) E)