Calculus

(I know it looks like a lot but it's just 10 questions)
(part 1 [It also explains my situation]: jiskha.com/questions/1788952/Hi-I-missed-school-due-to-illness-and-my-school-isnt-very-forgiving-so-I-would)

(If your willing to do this part along with part 1, you can take a break and do it tomorrow if you want, I don't want to force all this math on someone all at one, if there kind enough to help me)

1. What is the area bounded by y=x^2 and y=3x?
5
9/2
8
11.2
25

2. The region R is bounded by the axis, x = 2, and y = x^2. Which of these expressions represents the volume of the solid formed by revolving R about the line x = 2
(choices) (gyazo.com/80b96c5b657dbf0bbaca0c963534ea63)

3. Refer to the graph and information: An ant is crawling on a straight wire. The velocity, v(t), of the ant at time 0 <or= t <or= 8 is given in the graph. Note: The graph on 0 <or= r <or= 2 is a semi-circle (half-circle).
Given the following velocity curve, at which time (t) is the speed of the ant greatest?
(graph: gyazo.com/1b9c50ac2761b1f768297955677d1191)
0
2
3
4
8

4. An ants position during an 8 second time interval is shown by the graph below. What is the total distance the ant traveled over the time interval 2<=t<=8?
What is the total distance traveled by the ant over the time interval 2<=t<=8?
(graph: gyazo.com/28c7dc1f0f7e2a72fdff8a2dae66d6ec)
2
4
6
7
8

5. An ant is crawling on a straight wire. The velocity, v(t), of the ant at time 0<=t<=8 is given in the graph. Note: The graph on 0<=t<=2 is a semi-circle.
What is the total distance traveled by the ant over the time interval 0<=t<=8?
(Graph: gyazo.com/b9d87152987d355e01279955f8789410)
4-(pi/2)
4+(pi/2)
4+pi
(3pi/4)-5
7+(pi/2)

6. The average value of the function g(x)=3^cosx on the closed interval [-pi, 0] is:
30.980
18.068
7.593
4.347
1.325

7. Find the length of the arc defined by f(x) = (1/3)x^(3/2) on the interval from [0,5].
12.903
5.641
6.33
12.958
6.586

8. The temperature over a given period is:
(Chart: gyazo.com/d4c9f909d1f2787998e2ac771d99ca15)
Estimate the average temperature from 0<=t<=8 using the left endpoints of four equal subintervals
50° F
35° F
32.75° F
26° F
26.4° F

9. A solid has, as its base, the circular region in the xy-plane bounded by the graph of x^2+y^2=4. Find the volume of the solid if every cross section by a plane perpendicular to the x-axis is a quarter circle with one of it's radii in the base.
(8pi)/3
(128/5)pi
(64/3)pi
(32pi)/3
(32/5)pi

10. The change in the momentum of an object (Δ p) is fiven by the force, F, acting on the object multiplied by the time interval that the force was acting: Δ p = F Δ t.
If the force (in newtons) acting on a particular object is given by F(t) = cost, what's the total change in momentum of the object from time t=5 until t=7 seconds.
0.402 newton*sec
0.708 newton*sec
0.909 newton*sec
1.416 newton*sec
1.616 newton*sec

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asked by George
  1. #1. The curves intersect at (0,0) and (3,3) so the area is
    ∫[0,3] 3x - x^2 dx

    #2. using discs of thickness dy,
    v = ∫[0,4] πr^2 dy
    where r = 2-x = 2-√y
    v = ∫[0,4] π(2-√y)^2 dy
    Using shells of thickness dx,
    v = ∫[0,2] 2πrh dx
    where r=2-x and h=y=x^2
    v = ∫[0,2] 2π(2-x)x^2 dx

    #3. Surely you can read a graph ...

    #4. the distance traveled is just twice the sum of the heights of the two triangles. Think about it.

    #5. the distance is just the area of the semi-circle and the two triangles

    #6. as always, the average value is the area divided by the width:
    ∫[-π,0] 3^cosx dx / π ≈ 1.325

    #7. using the formula you have,
    s = ∫[0,5] √(1+(y')^2) dx = ∫[0,5] √(1+(√x/2)^2) dx = 1/4 ∫[0,5] √(x+4) dx

    #8. figure the area using 4 rectangles of width 2. Then divide by 8

    #9. It sounds like the quarter circle has radius y, so the area is 1/4 πy^2 = π/4 (4-x^2). Stacking up all those thin plates of thickness dx, and using the symmetry of the region,
    v = 2∫[0,2] π/4 (4-x^2) dx

    #10. you can approximate Δp = FΔt by
    dp = F dt
    p = ∫F dt
    over the interval of interest

  2. Could you please check these for me?

    1. 9/2
    2. I understand!
    3. Would it be 4?
    4. 3(2) + -1(2) = 4?
    5. 4+(pi/2)
    6. 1.325
    7. I got 3.16 but this is not an answer choice?
    8. 32.75
    9. 8pi/3
    10. This one is a bit confusing for me. How do I get the force?

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    posted by George

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