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course Mth 174
Section 7.81) 7.8.6 Does the integral Int((3x^2 - 6x + 1)/(4x^2 + 9) dx, 1, inf) diverge or converge and why?
Int((x^2-6x+1)/(x^2+4) dx, 1, inf) = int((x^2-6x+1)/(x^2+4) dx, 1, 5.828) + int((x^2-6x+1)/(x^2+4) dx, 5.828, inf)
On [ 5.828, inf]
0<=f(x)<=1/x^0
Int(1 dx,5.528, inf) is diverges, because p=0<1
Thus, int((x^2-6x+1)/(x^2+4) dx, 5.828, inf) is diverges
Therefore, Int((x^2-6x+1)/(x^2+4) dx, 1, inf) is diverges
@& You don't actually have to do the integral to prove convergence or divergence.
The value of (3x^2 - 6x + 1)/(4x^2 + 9) approaches 3/4 as x -> infinity. It follows that its integral from 0 to infinity diverges.
More formally:
Since f(x) approaches 3/4 as x -> infinity, for any value of epsilon there is a point beyond which | f(x) - 3/4 | < epsilon, i.e., a value N such that whenever x > N, the inequality holds.
Then the integral of f(x), from N to infinity, would diverge.
The integral of f(x) from 0 to N is finite, since f(x) is finite for all values of x (this would not be the case if the denominator was, say, 4 x^2 - 9, which could approach 0 for some value of x).
So the integral of f(x) from 0 to N is finitie, and from N to infinity diverges. Adding a finite quantity to a divergent quantity doesn't affect the divergence, so the integral from 0 to infinity diverges.*@
2) 7.8.12 Does the integral Int( 1/(x^4 + 2) dx, 1, inf) diverge or converge and why?
On [1, inf]
0<=1/(x^3+1)<=1/x^3
Int(1/x^3 dx, 1, inf) is converges, because p = 3 > 1
Thus, int(1/(x^3+1) dx, 1, inf) is converges
3) Does the integral Int(1/sqrt(x^2+1) dx, 1, inf) diverge or converge and why?
On [1, inf]
0<=1/(x^2+1)^(1/2)<1/x
Int(1/x dx, 1, inf) is diverges, because p=1
Thus, int(1/(x^2+1)^(1/2) dx, 1, inf) is diverges
@& Being less than a divergent quantity does not prove divergence. For example 1/x^2 < 1/x on this interval, but 1/x^2 converges.
You could easily show that on this interval, for example,
1 / sqrt( x^2 + 1 ) > 1/3 * 1/x,
which would then prove divergence.*@
4) 7.8.20 Does the integral Int(1/sqrt(x^3 + x) dx, 0, 1) diverge or converge and why?
On [0, 1]
0<=1/(x^3+x)^(1/2)<1/x^(3/2)
Int(1/x^(3/2) dx, 0, 1) is diverges, because p = 3/2 >1
Thus int(1/(x^3+x)^(1/2) dx, 0, 1) is diverges
@& This question is addressed in the Query, so you will respond to it there.*@
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Section 8.1
1) 8.1.4 Consider a circle with radius 3, and a horizontal strip with width `dy at position y above the origin.
• Write a Riemann sum for the reigon
• Write a definite integral representing the area of the reigon.
• Evaluate this integral exactly.
X^2 + y^2 = 9, x = `sqrt(9 - y^2)
DA = x dy = `sqrt(9 - y^2) dy
Riemann sum = sum(`sqrt(9 - y^2) dy)
A = int(dA) = 2int(`sqrt(9 - y^2) dy, y, 0, 3) = 4(h`sqrt(9-h^2)+9arcta(h/3) 0, 3 = 28.274
2) 8.1.6 Consider the area between the line y = h and the graph of y = |x| containing a horizontal strip with width `dh.
• Write a Riemann sum for the reigon
• Write a definite integral representing the area of the reigon.
• Evaluate this integral exactly.
Wi/2h = hi/h, wi = 2hi
A = sum(wi dh) = Sum(2hi dh)
A = int(2h dh 0,h) = (h^2 0, h) = h^2
3) 8.1.12 Consider the half disk with radius 7m and thickness 10m. `dy is parallel to the base.
• Write a Riemann sum for the reigon
• Write a definite integral representing the area of the reigon.
• Evaluate this integral exactly.
H^2 + (w/2)^2 = 49
W = 2sqrt(49-h^2)
A = SUM(10*2sqrt(49-h^2) dh)
A = int(10*2sqrt(49-h^2) dh 0, 7) = 20int(sqrt(49-h^2) dh 0, 7) = (490 arcsin(h/7) 0, 7) = 490arcsin1 = 796.690
4) 8.1.18 Consider the integral Int(7(1-h/3) dh, 0, 4).
• What shape does this integral represent?
• If it is a triangle what is its height or if it is a circle what is its radius?
• Make and describe a detailed sketch of the reigon which shows the variable and other relevent quantities.
It represent triangle.
Wi = 7/(1-h/3)
Wi/7 = (3-h)/3
Then, the height is 3 and the base is 4
5) 8.1.20 Consider the integral Int(pi(x/4)^2 dx, 0, 10).
• What shape does this integral represent?
• If it is a cone what is its height and radius or if it is a sphere what is its radius?
• Make and describe a detailed sketch of the reigon which shows the variable and other relevent quantities.
This represents a cone.
ri = wi/2 = (x/4)
which R = 10/4 = 2.5
Int(pi(x/4)^2 dx, 0, 10) = 20.8333333 `pi =pi HR^2 = H(100/16) pi
H = 10/3
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Section 8.2
1, 3, 6, 8, 11, 13, 16, 19, 20, 23, 26, 27, 31, 34, 37, 40, 41
1) 8.2.6 Find the volume of the reigon bounded by y = sin x, y = 0, x = 0, x = pi
V = int(pi(sin^2(x)) dx 0, pi) - int(0) = pi int(sin^2(x) dx 0, pi) = 4.935
2) 8.2.8 Find the volume of the reigon bounded by sqrt(cosh 2x) y = 0, x = 0, x = 2 rotated around the x-axis.
V = int(pi(sqrt(cosh 2x)^2 dx, 0, 2) = 1/2 int(sqrt(e^x + e^-x)^2 dx, 0, 2) = 1/2 e^x - e^-x 0, 2 =
@& you lost the pi at some point, but otherwise OK*@
3) 8.2.11 Find the arc length of the graph of y = sqrt(x^3) from x = 0 to x = 2.
Arc length = int(sqrt(1+(dy/dx)^2) 0, 2) = int(sqrt(1+(3x^2/sqrt(x^3))^2) 0, 2) = int( sqrt(1+9x) 0, 2) = 6.061
4) 8.2.16 Find the length of the parametric curves, x = sin(2t), y = cos(3t) for 0 <= t <= 2pi.
Length of parametric curve = int(sqrt((dx/dt)^2+(dy/dt)^2) 0, 2pi) = int(sqrt((2cos(2t))^2 + (-3sin(3t))^2) 0 , 2pi) = int( sqrt(4cos^2(2t)+9sin^2(3t) 0, 2pi) = 26.129
5) 8.2.20 Sketch the solid obtained by rotation the reigon y = sqrt(x), x = 4, y = 0 rotated around the line x = 4. Using your sketch describe how to approximate the volume of the solid by a Riemann sum and then find the volume.
R = (4-x), y =sqrt(x), then x = y^2
V = int(`pi(4-y^2)^2 dy, 0, sqrt(4)) = pi int(16 - 8y^2 + y^4 dy, 0, 2) = `pi(16y - 8y^3/3 + y^5/5 0, 2) = 53.6165
6) 8.2.26 Consider the region of the solid bounded by y = x^3, y = 1, and the y-axis. Find the area of the solid whose base is this reigon and whose cross-sections perpendicular to the x-axis are circles
R = (1-x^3)/2,
V = int(`pi((1-x^3)/2)^2 dx, 0, 1) = `pi/4 int(1-2x^3+x^6 dx, 0, 1) = `pi/4(x-x^4/2+x^7/7 0, 1) = 1.2903
7) 8.2.31 Consider the region bounded by y= e^x, the x-axis, and the lines x = 0 and x = 1. Find the area of the solid whose base is this reigon and the cross-sections perpendicular to the x-axis are squares.
L = e^x
V = int((e^x)^2 dx, 0,1) = int(e^(2x) dx, 0,1) = e^(2x)/2 0, 1) = 1/2 (e^2 - e^0) = 3.1945
8) 8.2.34 A 1m gutter is made of three strips of metal, each 7cm wide. The sides of the gutter form a trapezoid without a base where the angles between the pieces of metal are 120 degrees.
• Find the volume of water in the gutter when the depth of the water is h cm.
• What is the maximum value of h?
• What is the maximum volume of water the gutter can hold?
• If the gutter is holding half of the maximum amount of water, is the depth larger or smaller to the maximal h you found earlier?
• Find the depth of the water when the gutter is holding half of the maximum amount of water.
h/((wi-7)/2) = tan60
wi = (2sqrt(3) h)/3 + 7
V = int(100wi dh, 0 , 7sqrt(3)/2)
V(h) = 200sqrt(3)h^2/6 + 700h
Vmax = V(7sqrt) = 6365.286
V(half) = 6365.286/2 = 200sqrt(3)h^2/6 + 700h
h= 3.523 >7sqrt(3)/4, larger
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@& Very good, but you did leave pi out of a couple of the volume integrals.*@