Essential Knowledge 3.2A1

Essential Knowledge 3.2A1 Students will know that a Riemann sum, which requires the partition of an interval I, is the sum of products, each of which is the value of the function at a point in a subinterval multiplied by the length of that subinterval of the partition.

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Make sure you look at 3.2A2 before coming here.

Our estimations with rectangles were nice, but how can we calculate the exact answer, which would require an infinite number of rectangles? We're going to need to come up with some kind of formula to make that happen. Here, we'll start to develop that formula. Your goal is to understand what all of the different parts of the formula represent

We'll reuse the example from last time

Example

A car is rolling along the road at 5 ft/sec. Then the car begins accelerating such that the velocity is given by the equation:

v = 5t² + 5.

Determine the distance the car will have traveled 3 seconds after it begins accelerating.

OK, so when we made our 4 rectangles the first thing we did was divide the time into subintervals. To do that, we found the total time divided by the number of rectangles we wanted to make

(t_final - t_initial) / (# of rectangles) = (3 - 0) / 4 = 0.75 sec

The length of each subinterval is 0.75 seconds. From now on we'll use these letters instead.

(b - a) / n = Δx

Where

• b: end of interval

• a: beginning of interval

• n: number of subintervals (rectangles)

• Δx: length of a subinterval

Now let's label some parts of the interval. The left side of our first rectangle is at x = 0. Since this is the first x value we care about, let's call it x₁.

x₁ = 0

The left side of our next interval is x = 0.75. We'll call this x₂. We can continue this pattern

x₁ = 0

x₂ = 0.75

x₃ = 1.5

x₄ = 2.25

We used all these values last time, we just didn't give them special names.

After we found the lengths, we needed to find the height of the rectangles. In order to do that, we substituted the left endpoint of each interval into the velocity equation.

First rectangle

let t = 0

v = 5(0)² + 5 = 5 ft/sec

Second rectangle

let t = 0.75

v = 5(0.75)² + 5 = 7.8125 ft/sec

Third Rectangle

let t = 1.5

v = 5(1.5)² + 5 = 16.25 ft/sec

Fourth Rectangle

let t = 2.25

v = 5(1.5)² + 5 = 30.3125 ft/sec

Now, let's do it with our new notation. We'll treat the velocity equation as a function, which will be

f(x) = 5x² + 5

Substituting our left endpoints, we'd get the values

f(x₁) = 5 ft/sec

f(x₂) = 7.8125 ft/sec

f(x₃) = 16.25 ft/sec

f(x₄) = 30.3125 ft/sec

Then we found our area by multiplying the base x height for each rectangle, then adding them together.

Area = f(x₁)Δx₁ + f(x₂)Δx₂ + f(x₃)Δx₃ + f(x₄)Δx₄

(adding subscripts for Δx may feel unnecessary since they're all the same length, but it doesn't hurt and it will be useful later)

Well, we're starting to get a nice little formula here. This works for an approximation with 4 rectangles, but you can see the pattern here. For any number of rectangles, the pattern would be

Area = f(x₁)Δx₁ + f(x₂)Δx₂ + f(x₃)Δx₃ + ... + f(x_n)Δx_n

where n is the number of rectangles. This makes sense because if we have 25 rectangles, our final rectangle will be the 25th rectangle, and f(x_n)Δx_n would be f(x₂₅)Δx₂₅

Technically we've done it, but no one likes ... in their formula. When a formula sums up many terms, we like to use a summation symbol.

= f(x₁)Δx₁ + f(x₂)Δx₂ + f(x₃)Δx₃ + ... + f(x_n)Δx_n

And thus we've done it. This is the formula for creating n rectangles under a curve to approximate the area.

Area =