F1 – Definite Integrals - Area under Curves as Limits and Sums

IB Mathematics HL/SL & MCB4U

(A) Review

• We have looked at the process of anti-differentiation (given the derivative, can we find the “original” equation?)

• Then we introduced the indefinite integral which basically involved the same concept of finding an “original”equation since we could view the given equation as a derivative

• We introduced the integration symbol

• Now we will move onto a second type of integral the definite integral

(B) The Area Problem

• to introduce the second kind of integral : Definite Integrals we will take a look at “the Area Problem” the area problem is to definite integrals what the tangent and rate of change problems are to derivatives.

• The area problem will give us one of the interpretations of a definite integral and it will lead us to the definition of the definite integral.

(B) The Area Problem

• Let’s work with a simple quadratic function, f(x) = x2 + 2 and use a specific interval of [0,3]

• Now we wish to find the area under this curve

(C) The Area Problem – An Example

• To estimate the area under the curve, we will divide the are into simple rectangles as we can easily find the area of rectangles A = l × w

• Each rectangle will have a width of x which we calculate as (b – a)/n where b represents the higher bound on the area (i.e. x = 3) and a represents the lower bound on the area (i.e. x = 0) and n represents the number of rectangles we want to construct

• The height of each rectangle is then simply calculated using the function equation

• Then the total area (as an estimate) is determined as we sum the areas of the numerous rectangles we have created under the curve

• AT = A1 + A2 + A3 + ….. + An

• We can visualize the process on the next slide

(C) The Area Problem – An Example

• We have chosen to draw 6 rectangles on the interval [0,3]

• A1 = ½ × f(½) = 1.125

• A2 = ½ × f(1) = 1.5

• A3 = ½ × f(1½) = 2.125

• A4 = ½ × f(2) = 3

• A5 = ½ × f(2½) = 4.125

• A6 = ½ × f(3) = 5.5

• AT = 17.375 square units

• So our estimate is 17.375 which is obviously an overestimate

(C) The Area Problem – An Example

• In our previous slide, we used 6 rectangles which were constructed using a “right end point” (realize that both the use of 6 rectangles and the right end point are arbitrary!) in an increasing function like f(x) = x2 + 2 this creates an over-estimate of the area under the curve

• So let’s change from the right end point to the left end point and see what happens

(C) The Area Problem – An Example

• We have chosen to draw 6 rectangles on the interval [0,3]

• A1 = ½ × f(0) = 1

• A2 = ½ × f(½) = 1.125

• A3 = ½ × f(1) = 1.5

• A4 = ½ × f(1½) = 2.125

• A5 = ½ × f(2) = 3

• A6 = ½ × f(2½) = 4.125

• AT = 12.875 square units

• So our estimate is 12.875 which is obviously an under-estimate

(C) The Area Problem – An Example

• So our “left end point” method (now called a left hand Riemann sum) gives us an underestimate (in this example)

• Our “right end point” method (now called a right handed Riemann sum) gives us an overestimate (in this example)

• We can adjust our strategy in a variety of ways one is by adjusting the “end point” why not simply use a “midpoint” in each interval and get a mix of over- and under-estimates? see next slide

(C) The Area Problem – An Example

• We have chosen to draw 6 rectangles on the interval [0,3]

• A1 = ½ × f(¼) = 1.03125• A2 = ½ × f (¾) = 1.28125• A3 = ½ × f(1¼) = 1.78125• A4 = ½ × f(1¾) = 2.53125• A5 = ½ × f(2¼) = 3.53125• A6 = ½ × f(2¾) = 4.78125• AT = 14.9375 square units

which is a more accurate estimate (15 is the exact answer)

(C) The Area Problem – An Example

• We have chosen to draw 6 trapezoids on the interval [0,3]

• A1 = ½ × ½[f(0) + f(½)] = 1.0625• A2 = ½ × ½[f(½) + f(1)] = 1.3125• A3 = ½ × ½[f(1) + f(1½)] = 1.8125• A4 = ½ × ½[f(1½) + f(2)] = 2.5625• A5 = ½ × ½[f(0) + f(½)] = 3.5625• A6 = ½ × ½[f(0) + f(½)] = 4.8125• AT = 15.125 square units

• (15 is the exact answer)

(D) The Area Problem – Expanding our Example

• Now back to our left and right Riemann sums and our original example how can we increase the accuracy of our estimate?

• We simply increase the number of rectangles that we construct under the curve

• Initially we chose 6, now let’s choose a few more … say 12, 60, and 300 ….

(D) The Area Problem – Expanding our Example

# of rectangles Area estimate

12 16.15625

60 15.22625

300 15.04505

(D) The Area Problem – Expanding our Example

# of rectangles Area estimate

12 13.90625

60 14.77625

300 14.95505

(E) The Area Problem - Conclusion

• We have seen the following general formula used in the preceding examples:

• A = f(x1)x + f(x2)x + …. + f(xi)x + ….. + f(xn)x as we have created n rectangles

• Since this represents a sum, we can use summation notation to re-express this formula

• So this is the formula for our Riemann sum

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(F) Riemann Sums – Internet Interactive Example

• Visual Calculus - Riemann Sums

• And some further worked examples showing both a graphic and algebraic representation:

• Visual Calculus - Riemann Sums – 2

(G) The Area Problem – Further Examples

• (i) Find the area between the curve f(x) = x3 – 5x2 + 6x + 5 and the x-axis on [0,4] using 5 intervals and using right- and left- and midpoint Riemann sums. Verify with technology.

• (ii) Find the area between the curve f(x) = x2 – 4 and the x-axis on [0,2] using 8 intervals and using right- and left- and midpoint Riemann sums. Verify with technology.

• (iii) Find the area between the curve f(x) = x2 – 2 and the x-axis on [0,2] using 8 intervals and using midpoint Riemann sums. Verify with technology.

(G) The Area Problem – Further Examples

• Homework to reinforce these concepts:

• Stewart, 1989, Chap 10.4, p474, Q3c,4d,5

• Stewart, 1989, Chap 10.5, p482, Q1,3

(H) The Area Problem – Exact Areas

• Now to make our estimate more accurate, we simply made more rectangles how many more though? why not an infinite amount (use the limit concept as we did with our tangent/secant issue in differential calculus!)

• Then we arrive at the following “formula”:

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(H) The Area Problem – Areas as Limits

• The formula we will use is

• The function will be f(x) = x3 – x2 - 2x - 1 on [0,2]

• Which we can graph and see

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(H) The Area Problem – Areas as Limits

• So our formula is:

• Now we need to work out just what f(xi) and x are equal to so we can sub them into our formula:

x = (b-a)/n = (2-0)/n = 2/n

• xi simply refers to the any endpoint on any one of the many rectangles so let’s work with the ith endpoint on the ith rectangle so in general, xi = a + ix

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(H) The Area Problem – Areas as Limits

• I am using a + idx on the diagram to represent the ith rectangle, which is ix (or idx) units away from a

• Then the height of this rectangle is f(a + ix)

(H) The Area Problem – Areas as Limits• Now back to the formula in which x = 2/n and xi = 0 + i x or xi = 2i/n

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(H) The Area Problem – Sums of Powers

• We have the following summation formulas to use in our simplification:

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(H) The Area Problem – Areas as Limits

• Now we simply substitute our appropriate power sums:

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(H) The Area Problem – Areas as Limits

• Now we can confirm this visually using graphing software (WINPLOT):

• And we get the total area between the axis and the curve to be -4.666666666 from the software!

(H) The Area Problem – Areas as Limits – Additional Examples

• (i) Find the area between the x-axis and the function f(x) = x2 on [0,2] using the areas as limits approach. Confirm with graphing technology

• (ii) Find the area between the x-axis and the function f(x) = x3 + x on [1,4] using the areas as limits approach. Confirm with graphing technology

(I) Areas as Limits - Homework

• Homework to reinforce these concepts:

• Stewart, 1989, Chap 10.4, p474, Q3,4,6*

Internet Links

• Calculus I (Math 2413) - Integrals - Area Problem from Paul Dawkins

• Integration Concepts from Visual Calculus

• Areas and Riemann Sums from P.K. Ving - Calculus I - Problems and Solutions