Chapter 2

Definite Integration

2.1 Definition of a Definite Integral

Let be defined (i.e. bounded) in the interval . Sub Divide I into n subintervals by the points

and let

, r = 1, 2, . . . ,n.

For each r choose a point such that . Then the sum

,called a Riemann sum.

If the limit of the sum as such that max exists and is independent of the choice of and , then we say that is (Riemann) integrable and the limit is written as

called the definite integral of over .In other words

.

Note that the above limit exist if is bounded in

Suppose on . Then the Riemann sum

,

is the sum of the areas of the n rectangles shown below, and thus represents an approximation to the area A under the graph on . Figure below illustrates the case where n = 5.

x

Thus provided is non-negative on , then

is the area enclosed by the curve , the ordinates and and the x-axis.

If the function is always negative on the negative value of this Riemann sum gives an approximation to the area between the graph of , the ordinates and and the x-axis.

2.2The Fundamental Theorem of Calculus

Calculating the value of a definite integral using Riemann sums is extremely difficult for all but simple functions. However it turns out that the definite integral is related to an indefinite integral of .

Theorem: If is a continuous function on and is an indefinite of then

[For proof of this important result consult any books on higher Calculus]

Exercise 2.2:

1. For the function defined by f(x) = 1 + 2x on [0, 4], calculate the Riemann sum corresponding to the partition 0 < 1 < 1.5 < 2 < 2.5 < 4 and the choice of points c1 = 0.5, c2 = 1, c3 = 2, c4 = 2.25, c5 = 3. 2. Estimate the value of the Riemann sum of the following integrals using n subintervals of equal length and choosing points ck as (i) the left endpoints of the subintervals, (ii) the right endpoints of the subintervals, (iii) the midpoints of the subintervals:

(a) ( n = 4) (b) ( n = 6)

(c) ( n = 4) (d) ( n = 6)

(e) ( n = 4) (f) ( n = 4)

3. Estimate the value of the above integrals of Question No. 2 using Trapezoidal rule.

4. State with reasons whether the following integrals exist or not.

(a) (b) (c) (d)

(e) (f) (g) (h)

5. Evaluate the following integrals.

(a) (b) (c)

(d) (e) (f) (g)

(h) (i) (j)

6. State whether the following functions are odd, even or neither.

(a) (b) (c)

(d) (e) (f)

7. Find the values of the following integrals:

(a) (b) (c)

(d) (e) (f) (g)

Answers:

1. 20 2. (a) 11.89 (b) 114 (c) 26.79 (d) 0.394 (e) 2 (f) 4 5. (a) 160.2 (b) 2 (c) Pi/12 (d) 2.58 (e) 0.89 (f) .161 (g) 135.7 (h) 0.5 (i) 4.57 (j) -0.39 6. (a) Odd (b) Even (c) Odd (d) Odd (e) neither Odd nor Even (f) Odd7. (a) 0 (b) 5.33 (c) 42.67 (d) 0 (e) -12.56 (f) 6.28 (g) 0

2.3Applications of the Definite Integral

2.3.1Areas of Regions Between Two Curves

Definite integrals could be used to determine the area of the region between the graph of a function and the x-axis.Recall that:

if for then the area of the region bounded by the curve , the x-axis and the lines and is .

if for then the area of the region bounded by the curve , the x-axis and the lines and is .In either case the area A is given by

A = .

We shall now turn our attention to the more general problem of finding the area of regions bounded by two curves. In this case the functions and are such that for all .

In the case where both functions are positive i.e. , and , then

Area = =

If , and , then

Area = =

In general, the area A between the two curves and in is

Here we must subtract from over the subintervals where and reverse the signs where the inequality is reversed.

When setting up integrals which give the area of regions between curves it is not always the most convenient to subdivide the region into vertical elements (i.e. elements parallel to the y-axis) but may be simpler to slice into horizontal strips.

Exercise 2.3.1:

1. Find the area of the shaded region.

2. Sketch the region enclosed by the curves and find its area.

(a) and the x-axis(b) and the x-axis

(c) and the y-axis(d) and the y-axis

3. Sketch the region enclosed by the curves and find its area.

(a) and (b) and

(c) (d)

(e) (f)

(g) (h)

4. Find the area bounded by the ellipse

5. Find the area between the parabolas and

6. Find the area between the parabolas and

7. Find the area bounded by the curve and the x-axis.

8. Find the area bounded by the curve and the x-axis.

9. Find the area bounded by the curve and the y-axis.

Answers:

1. (a) 4.5 (b) 22/3 (c) 1 (d) 10/3 2. (a) 36 (b) 32/3 (c) 36 (d) 32/33. (a) 4.5 (b) 4.5 (c) 49/192 (d) -0.5 (e) 4.5 (f) 0.5 (g) 1 (h) 93/8 4. ab 5. 16/3 6. 64/3 7. 253/12 8. 0.5 9. 0.5

2.3.2 Solids of Revolution:

1. Find the volume of the solid when the region enclosed by the given curves is revolved about x-axis.

(a)

(b)

(c)

(d)

(e)

2. Find the volume of the solid when the region enclosed by the given curves is revolved about y-axis.

(a)

(b)

(c)

3. Find the volume of solid generated by revolving the ellipse about x-axis.

4. Find the volume of solid generated by revolving the ellipse about y-axis.

Answers: 1. (a) 32/5 (b) 1296/5 (c) 4 (d) 2/35 (e) 2048/15 2. (a) 3/5 (b) 16/15 (c) 4 3. 8/3 4. 16/3

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