5128_Ch07_pp378-433

416 Chapter 7 Applications of Definite Integrals
Section 7.4 Exercises
In Exercises 1–10,
(a) set up an integral for the length of the curve;
(b) graph the curve to see what it looks like;
(c) use NINT to find the length of the curve.
1. y x 2 , 1 x 2
2. y tan x, p3 x 0
3. x sin y, 0 y p
4. x 1 y , 2 12 y 12
5. y 2 2y 2x 1, from 1, 1 to 7, 3
6. y sin x x cos x, 0 x p
x
7. y 0 tan tdt, 0 x p6
8. x y 0
sec 2 t 1 dt, p3 y p4
9. y sec x, p3 x p3
10. y e x e x 2, 3 x 3
In Exercises 11–18, find the exact length of the curve analytically by
antidifferentiation. You will need to simplify the integrand
algebraically before finding an antiderivative.
11. y 13x 2 2 32 from x 0 to x 3 12
12. y x 32 from x 0 to x 4 (8010 8)/27
13. x y 3 3 14y from y 1 to y 3
[Hint: 1 dxdy 2 is a perfect square.] 53/6
14. x y 4 4 18y 2 from y 1 to y 2
[Hint: 1 dxdy 2 is a perfect square.] 123/32
15. x y 3 6 12y from y 1 to y 2
[Hint: 1 dxdy 2 is a perfect square.] 17/12
16. y x 3 3 x 2 x 14x 4, 0 x 2 53/6
17. x y 0
sec 4 t 1 dt, p4 y p4 2
18. y x
3t 4 1 dt, 2 x 1 73/3
2
19. (a) Group Activity Find a curve through the point 1, 1
whose length integral is y x
L 4
1 1 4 x dx.
1
(b) Writing to Learn How many such curves are there? Give
reasons for your answer.
20. (a) Group Activity Find a curve through the point 0, 1
whose length integral is y 1/(1 x)
L 2
1 y 1 4 dy.
1
(b) Writing to Learn How many such curves are there? Give
reasons for your answer.
21. Find the length of the curve
y x
from x 0 to x p4. 1
0
cos2t dt
22. The Length of an Astroid The graph of the equation
x 23 y 23 1 is one of the family of curves called astroids
(not “asteroids”) because of their starlike appearance (see figure).
Find the length of this particular astroid by finding the length of
half the first quadrant portion, y 1 x 23 32 , 24 x 1,
and multiplying by 8. 6
y
23. Fabricating Metal Sheets Your metal fabrication company is
bidding for a contract to make sheets of corrugated steel roofing
like the one shown here. The cross sections of the corrugated
sheets are to conform to the curve
y sin ( 3 p
20 ) x , 0 x 20 in.
If the roofing is to be stamped from flat sheets by a process that
does not stretch the material, how wide should the original
material be? Give your answer to two decimal places. 21.07 inches
Original sheet
–1
20 in.
Corrugated sheet
24. Tunnel Construction Your engineering firm is bidding for
the contract to construct the tunnel shown on the next page. The
tunnel is 300 ft long and 50 ft wide at the base. The cross
section is shaped like one arch of the curve y 25 cos px50.
Upon completion, the tunnel’s inside surface (excluding the
roadway) will be treated with a waterproof sealer that costs
$1.75 per square foot to apply. How much will it cost to apply
the sealer? $38,422
1
0
–1
O
y
x 2/3 y 2/3 1
y sin —–
3
x
20
20
1
x
x (in.)
19. (b) Only one. We know the derivative of the function and the value of the
function at one value of x.
20. (b) Only one. We know the derivative of the function and the value of the
function at one value of x.