PRACTICE4 Name Section
1. Determine the work to empty the water from a filled semi-spherical swimming pool with radius 10 ft.
x
−10
0
0 10
−5
−10 F y
= 62 .
4 π (
√
100 − y 2 ) 2 ∆ y lbs
D y
= − y ft
W =
Z
0
62 .
4 π ( y 3
− 100 y ) dy
−
10
= 62 .
4 π
1
4 y 4
−
100
2 y
= 62 .
4 π (2500) ft-lbs.
2
0
−
10
2. A cable that weighs 2 lb/ft is used to lift 800 lb of coal up a mine shaft 500 ft deep.
Find the work done.
W coal = 800(500) ft-lbs = 400
, 000 ft-lb
Z
500
W rope = 2(500 − y ) dy
0
500
= 2 500 y −
1
2 y
= 250 , 000 ft-lbs
2
0
Work=650,000 ft-lb
3. A spring has natural length 20 cm. If a 25-N force is required to keep it stretched to a length of 30 cm (determining k, the spring constant), how much work is required to stretch it from 20 cm to 25 cm?
F = kx ⇒ 25 = k ( .
3) ⇒ k = 250 / 3
W =
=
=
Z
1 / 4
1 / 5
250 x dx
3
250 x 2
3
250
3
1 / 4
1
1 / 5
16
−
1
25
J

4. SET UP (do not evaluate) the integral to determine the force on the trapezoidal side of the tank filled with water that has a rectangular bottom with length 12 ft and width 10 ft and rectanular top with length 20 ft and width 10 ft. The tank is
8 ft deep from bottom to top.
−10 −5
11
10
9
8
7
6
5
4
3
2
1
0
0 15 5 x
10
Equation of the line having points (6 , 0) and (10 , 8) is y = 2 x − 12 y + 2
A y
= (2) ∆ y
P y
2
= 62 .
4(8 − y )
Z
8
F = 62 .
4(8 − y )( y + 12) dy
0
5. SET UP (do not evaluate) the integral to determine the work to empty this trapezoidal tank.
F y
= 62 .
4(10)(2)
D y
= (8 − y ) y + 2
2
∆ y
W =
Z
8
0
624(8 − y )( y + 12) dy
6. Determine the arclength of the curve f ( x ) = f ′ ( x ) = x −
4
1 x
So f ′ ( x ) 2 = x 2
−
1
2
+ x 2
2
1
−
16 x 2
.
ln x
4 for 1 ≤ x ≤ 4.
A.L.
=
=
=
Z
4
1 s
1 + x 2 −
1
2
Z
4
1
1
2 x 2 s
1 x +
4 x
+
1
4 ln x
+
1
16 x 2 dx =
Z
4
1
4
1
2 dx =
= 8 +
Z
4
1
1
4
1 x +
4 x
1 ln 4 −
2 s x 2 + dx
1
2
+
1
16 x 2 dx