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AP Calculus BC 5.2 Definite Integrals

Objective: able to express the area under a curve as a definite integral and as a limit of Riemann

sums; to compute the area under a curve using a numerical integration procedure.

Summation Notation

Use Sigma notation to write the sum of:

1. a1 2 n 2. 1st 10 Whole Numbers 3. 1

st

4. 1st 20 Odd Numbers 5. 3(1)

2 + 3(2)

2 +…+3(100)

26.

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Consider an arbitrary continuous function f(x) defined on a closed interval [a, b]. It may have

negative values as well as positive values.

Partition the interval [a, b] into n subintervals where

a = x0 < x1 < x2 < … < xk-1 < xk < … < xn-1 < xn = b.

This partition determines n closed subintervals.

The kth

subinterval is [xk-1, xk], which has length Δxk = xk – xk-1 .

Within each subinterval, select some number. We write the number chosen from the kth

subinterval as ck.

On each subinterval, stand a vertical rectangle reaching from the x-axis to touch the curve at

(ck, f(ck)). These rectangles can lie either above or below the x-axis.

On each subinterval, calculate the area, (height)(base), of the rectangle using f(ck)∙Δ xk. This

product can be positive, negative, or zero, depending on f (ck).

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Finally, take the a, b] of these products:

Think of the pages in a textbook and the side area

The Definite Integral as a Limit of Riemann Sums

The Existence of Definite Integrals

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The Definite Integral of a Continuous Function on [a, b]

is used to represent

7.

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Area Under a Curve (as a Definite Integral)

8. Evaluate the integral by using the graph of the integrand (use your knowledge of area).

Now find the area 'under' the curve.a) b)

Area Under a Curve Part 2 (as a Definite Integral)

If y = f(x f

over an interval [a, b] are negatives of rectangle areas: Area = when f(x) ≤ 0

9. Evaluate the integral by using the graph of the integrand (use your knowledge of area).

Now find the area 'under' the curve.a)

b)

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Area Under a Curve Part 3 (as a Definite Integral)

If y = f(x) is integrable and has both positive and negative values, then

= (area above the x-axis) - (area below the x-axis) Now find the area 'under' the curve.

The Integral of a Constant Function

11. Find the distance traveled by a train moving at a constant speed of 80 mph from 6am to

10:30am.

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I don't get it at allpretty well