1)What is the area bounded by y = x^2 and y =3x?
A)5
B)9/2
***C)8
D)11.2
E)25
i believe it to be 8 but im not sure.
2)The region R is bounded by the x-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?
I believe it to be the Integral from 0 to 4( pi(2- sqr root of y)^2
dy
3)Refer to the graph and information: An ant's 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 0 ≤ t ≤ 8?
I don't even know what to do with this one
4)The average value of the function g(x) = 3^cos(x) on the closed interval [ − �€ , 0 ] is:
***A)30.980 My answer. Not sure :(
B)18.068
C)7.953
D)4.347
E)1.325
5) Find the length of the arc defined by f(x)= (1/3x^3/2) on the interval from [0,5].
A)12.903
B)5.641
C)6.333
D)12.958
E)6.586
NOT sure
6)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 its radii in the base.
PLEASE HELP ME with what you can. You dont have to answer all but help me with the easy short ones PLEASE. My test in close date and this is a big thing for me. Thank you :)
#1 a = ∫[0,3] (3x-x^2) dx = 9/2
#2 Using either discs or shells,
v = ∫[0,4] π(2-x)^2 dy
= ∫[0,4] π(2-√y)^2 dy = 8π/3
or
v = ∫[0,2] 2π(2-x)y dx
= ∫[0,2] 2π(2-x)x^2 dx = 8π/3
#3 I can't make out the function, but the total distance traveled is the arc length of the path followed:
∫√(1+y'^2) dx
over the appropriate interval.
Depending on the graph, it might be easier just to do a geometric evaluation. But since this is a calculus test, I suppose not.
#4 Again, copy/paste mangling, but the average value of f(x) on [a,b] is
(∫[a,b] f(d) dx)/(b-a)
Since 3^cosx is not directly integrable, I suspect an algebraic shortcut is being used, or a numeric approximation is wanted.
#5 f(x) = 1/3 x^3/2
f' = 1/2 x^1/2
s = ∫[0,5] √(1+x/4) dx = 19/3
#6 If the radius is a chord of the circle, then r = 2y = 2√(4-x^2)
Thus, adding up all those slices, each of area πr^2/4, and taking advantage of symmetry,
v = 2∫[0,2] π(4-x^2) dx = 32π/3
1) To find the area bounded by y = x^2 and y = 3x, you need to find the points of intersection first. Set the two equations equal to each other and solve for x:
x^2 = 3x
Rearranging, you get x^2 - 3x = 0. Factoring out x, you have x(x - 3) = 0. So, x = 0 or x = 3.
These are the points of intersection. To find the area, you need to integrate the difference between the two functions over the interval [0, 3]. So, the area is given by the integral:
∫[0,3] (3x - x^2) dx.
Evaluating this integral will give you the area bounded by the two curves. In this case, the integral evaluates to 8.
Therefore, the answer is C) 8.
2) To find the volume of the solid formed by revolving the region R about the line x = 2, you need to use the method of cylindrical shells. The volume of each cylindrical shell is given by the formula 2πrhΔx, where r is the distance from the axis of rotation, h is the height of the shell, and Δx is the infinitesimal thickness of the shell.
In this case, the radius r is equal to 2 - x, and the height h is equal to x^2. The limits of integration are from x = 0 to x = 2. Therefore, the integral expression for the volume is:
V = ∫[0,2] 2π(x^2)(2 - x) dx.
Evaluating this integral will give you the volume of the solid.
3) Without the graph or any specific information about the ant's position, it is difficult to determine the total distance traveled by the ant. Please provide more details or the graph for further assistance.
4) To find the average value of the function g(x) = 3^cos(x) on the closed interval [−π, 0], you need to evaluate the definite integral of g(x) over that interval and divide it by the length of the interval (in this case, π).
The average value (AV) is given by the formula:
AV = (1/π) ∫[−π,0] g(x) dx.
To find the average value, evaluate this integral and divide it by π.
5) To find the length of the arc defined by f(x) = (1/3)x^(3/2) on the interval [0,5], you need to use the formula for arc length.
The formula for arc length of a function f(x) on an interval [a,b] is given by:
L = ∫[a,b] √(1 + (f'(x))^2) dx.
In this case, you need to find the derivative of f(x) and substitute it into the formula. The result is the integral:
L = ∫[0,5] √(1 + (2/3)x^(1/2))^2 dx.
Evaluating this integral will give you the length of the arc.
6) To find the volume of the solid with a circular base bounded by x^2 + y^2 = 4, where each cross-section is a quarter circle with one radius in the base, you can use the method of slicing.
Since each cross-section is a quarter circle, the radius will vary along the x-axis. The radius at any given x-coordinate is the y-coordinate of the circle on the x-axis. Therefore, the radius is √(4 - x^2).
To find the volume, you integrate the area of each cross-section over the interval [-2, 2]. The integral expression for the volume is:
V = ∫[-2,2] (1/4)π(√(4 - x^2))^2 dx.
Evaluate this integral to find the volume of the solid.
Best of luck with your test!