1. (a) By reading values from the given graph of f, use four rectangles to find

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Question

(a) By reading values from the given graph of f, use four rectangles to find a lower estimate and an upper estimate for the area under the given graph f from x=0 to x=8. In each case sketch the rectangles that you use.

(b) Find new estimates using eight rectangels in each case.

(a) Estimate the area under the graph of f(x)=1+x^2 from x=-1 to x=2 using three rectangles and right endpoints. Then improve your estimate by using six rectanlges. Sketch the curve and the approximating rectangles.

(b) Repeat part (a) using left endpoints.

(d) From you sketches in parts (a)-(c), which appears to be the best estimate?

PS. PLEASE show me STEP BY STEP. BECAUSE I WANT TO TRY MYSELF ALSO NEED HELP REALLY.

By reading values from the given graph of f, use four rectangles to find a lower estimate and an upper estimate for the area under the given graph f from x=0 to x=8. In each case sketch the rectangles that you use. Find new estimates using eight rectangels in each case.

Explanation / Answer

GEXERCISES

I. (a) By reading values from the given graph of f, use five rectangles

to find a lower estimate and an upper estimate for

the area under the given graph of f from x = 0 to x = 10.

In each case sketch the rectangles that you use.

(b) Find new estimates using ten rectangles in each case.

.......-

5 i/ y=f(x)

0 5 10 x

[b] (a) Use six rectangles to find estimates of each type for the

area under the given graph of f from x = 0 to x = 12.

(i) L6 (sample points are left endpoints)

(ii) R6 (sample points are right endpoints)

(iii) M6 (sample points are midpoints)

(b) Is L6 an underestimate or overestimate of the true area?

(d) Which of the numbers L6, R6, or M6 gives the best

estimate? Explain.

r----- 8 '-.

"---..

y=f(x) "'"

"" "

0 4 8 12 x

3. (a) Estimate the area under the graph of f(x) = cos x from

x = 0 to x = Tr/2 using four approximating rectangles

and right endpoints. Sketch the graph and the rectangles.

Is your estimate an underestimate or an overestimate?

(b) Repeat part (a) using left endpoints.

4. (a) Estimate the area under the graph of f(x) = J-; from

x = 0 to x = 4 using four approximating rectangles and

right endpoints. Sketch the graph and the rectangles. Is

your estimate an underestimate or an overestimate?

(b) Repeat part (a) using left endpoints.

[I] (a) Estimate the area under the graph of f(x) = 1 + x2 from

x = - I to x = 2 using three rectangles and right endpoints.

Then improve your estimate by using six rectangles.

Sketch the curve and the approximating rectangles.

(b) Repeat part (a) using left endpoints.

(d) From your sketches in parts (a)-(c), which appears to

be the best estimate?

~ 6. (a) Graph the function f(x) = 1/(1 + x2), -2 ,-;;x ,-;; 2.

(b) Estimate the area under the graph of f using four approximating

rectangles and taking the sample points to be

(i) right endpoints (ii) midpoints

In each case sketch the curve and the rectangles.

rectangles.

7-8 With a programmable calculator (or a computer), it is possible

to evaluate the expressions for the sums of areas of approximating

rectangles, even for large values of n, using looping. (On a TI use

the Is> command or a For-EndFor loop, on a Casio use Isz, on an

HP or in BASIC use a FOR-NEXT loop.) Compute the sum of the

areas of approximating rectangles using equal subintervals and

right endpoints for n = 10, 30, 50, and 100. Then guess the value

of the exact area.

7. The region under y = x4 from 0 to 1

8. The region under y = cos x from 0 to Tr/2

[ill] 9. Some computer algebra systems have commands that will

draw approximating rectangles and evaluate the sums of their

areas, at least if xi' is a left or right endpoint. (For instance,

in Maple use leftbox, rightbox, leftsuffi, and

rightsum.)

(a) If f(x) = 1/(x2 + 1),0'-;; x'-;; 1, find the left and right

sums for n = 10, 30, and 50.

(b) Illustrate by graphing the rectangles in part (a).

and 0.791.

[ill] 10. (a) Iff(x) = x/(x + 2), I ,-;; x'-;; 4, use the commands

discussed in Exercise 9 to find the left and right sums for

n = 10, 30, and 50.

(b) Illustrate by graphing the rectangles in part (a).

and 1.624.

[TIJ The speed of a runner increased steadily during the first three

seconds of a race. Her speed at half-second intervals is given in

the table. Find lower and upper estimates for the distance that

she traveled during these three seconds.

t (s) 0 0.5 1.0 1.5 2.0 2.5 3.0

v (m/s) 0 1.9 3.3 4.5 5.5 5.9 6.2

12. Speedometer readings for a motorcycle at l2-second intervals

are given in the table.

(a) Estimate the distance traveled by the motorcycle during

this time period using the velocities at the beginning of

the time intervals.

(b) Give another estimate using the velocities at the end of

the time periods.

estimates? Explain.

t (s) 0 12 24 36 48 60

v (m/s) 9.1 8.5 7.6 6.7 7.3 8.2

13. Oil leaked from a tank at a rate of ref) liters per hour. The

rate decreased as time passed and values of the rate at twohour

time intervals are shown in the table. Find lower and

upper estimates for the total amount of oil that leaked out.

t (h) 0 2 4 6 8 10

r(t) (L/h) 8.7 7.6 6.8 6.2 5.7 5.3

14. When we estimate distances from velocity data, it is sometimes

necessary to use times to, t), f2, f3, ... that are not

equally spaced. We can still estimate distances using the time

periods t::.ti = ti - fi-I. For example, on May 7, 1992, the

space shuttle Endeavour was launched on mission STS-49,

the purpose of which was to install a new perigee kick motor

in an Intelsat communications satellite. The table, provided

by NASA, gives the velocity data for the shuttle between

liftoff and the jettisoning of the solid rocket boosters. Use

these data to estimate the height above the earth's surface of

the Endeavour, 62 seconds after liftoff.

Event Time (s) Velocity (m/s)

Launch 0 0

Begin roll maneuver 10 56

End roll maneuver 15 97

Throttle to 89% 20 136

Throttle to 67% 32 226

Throttle to 104% 59 404

Maximum dynamic pressure 62 440

Solid rocket booster separation 125 1265

15. The velocity graph of a braking car is shown. Use it to estimate

the distance traveled by the car while the brakes are

applied.

(m/s)

6 t

(seconds)

16. The velocity graph of a car accelerating from rest to a speed

of 120 km/h over a period of 30 seconds is shown. Estimate

the distance traveled during this period.

Ih) -- -----

80 /

40 /

0 10 20 30 t

seco

17-19 Use Definition 2 to find an expression for the area under

the graph of f as a limit. Do not evaluate the limit.

17. f(x) = $, I ~ x ~ 16

18. f(x) = I + x4, 2 ~ x ~ 5

19. f(x) = xcosx, 0 ~ x ~ 7T/2

20-21 Determine a region whose area is equal to the given limit.

Do not evaluate the limit.

" 2 ( 2i)IO 20. lim 2: - 5 + -

1/---+00 i=1 n n

Il 7T i7T [Q lim 2: - tan-

II--HtJ ;=1 4n 4n

22. (a) Use Definition 2 to find an expression for the area under

the curve y = x3 from 0 to 1 as a limit.

(b) The following formula for the sum of the cubes of the

first n integers is proved in Appendix E. Use it to evaluate

the limit in part (a).

[

n(n+I)]2 13 + 23 + 33 + ... + n3 = 2

ITill23. (a) Express the area under the curve y = x5 from 0 to 2 as

a limit.

(b) Use a computer algebra system to find the sum in your

expression from part (a).

ITill24. (a) Express the area under the curve y = x4 + 5x2 + x from

2 to 7 as a limit.

(b) Use a computer algebra system to evaluate the sum in

part (a).

evaluating the limit of the expression in part (b).

IT&J 25. Find the exact area under the cosine curve y = cos x from

x = 0 to x = b, where 0 .; b .; 71/2. (Use a computer algebra

system both to evaluate the sum and compute the limit.)

In particular, what is the area if b = 7T/2?

26. (a) Let All be the area of a polygon with n equal sides

inscribed in a circle with radius r. By dividing the polygon

into n congruent triangles with central angle 27T/n, show

that

=================~ THE DEFINITE INTEGRAL

We saw in Section 5.1 that a limit of the form

II OJ lim L f(xT) Lh = lim [J(xt) Llx +f(xi) Llx + ... +f(x,f) Llx]

n----"'OO ;= I n~OO

arises when we compute an area. We also saw that it arises when we try to find the distance

traveled by an object. It turns out that this same type of limit occurs in a wide variety

of situations even when f is not necessarily a positive function. In Chapters 6 and 9 we

will see that limits of the form (1) also arise in finding lengths of curves, volumes of solids,

centers of mass, force due to water pressure, and work, as well as other quantities. We

therefore give this type of limit a special name and notation.

[I] DEFINITION OF A DEFINITE INTEGRAL If f is a function defined for a ::;; x ::;; b,

we divide the interval [a, b] into n subintervals of equal width Llx = (b - a)/n.

We let Xo (= a), Xl, X2,

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