course PHY 174 dRYwfŎassignment #008
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15:30:12 query explain the convergence or divergence of series (no summary needed)
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RESPONSE --> A function converges if the area underneath approaches a finite value. A function diverges if the are underneath approaches infinity. ** An integral converges if the limit of its Riemann sums can be found and is finite. Roughly this corresponds to the area under the curve (positive for area above and negative for area below the axis) being finite and well-defined (the area under the graph of the sine function, from 0 to infinity, could be regarded as finite but it keeps fluctuating as the sine goes positive then negative, so it isn't well-defined). **
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16:47:51 explain how we know that the integral from 1 to infinity of 1 / x^p converges if p > 1, diverges for p < 1, diverges for p = 1
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RESPONSE --> For values of p less than 1, the function grows smaller as x gets larger, but at a rate that does not approach a finite value for the function as x gets arbitrarily large. For p = 1, the function grows smaller as x gets larger, approaching zero. For all p greater than 1, the function gets smaller at a greater rate, approaching zero.
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19:56:11 explain how we know that the integral from 0 to 1 of 1 / x^p diverges if p > 1, converges for p < 1, diverges for p = 1
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RESPONSE --> diverges for p >= 1 converges for p < 1 For p > 1 integral of 1/x^2 from 0 to 1 as x gets arbitrarily small, the integral does not approach a finite number. For p = 1 integral of 1/x from 0 to 1 as x gets arbitrarily small, the integral does not approach a finite number. For p = .5 integral of 1/x^.5 from 0 to 1 as x gets arbitrarily small, the integral approaches 2.
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20:28:27 explain how we know that the integral from 0 to infinity of e^(-a x) converges for a > 0.
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RESPONSE --> For positive values of a, as x approaches infinity, the function does not approach a finite value. For negative values of a, as x approaches infinity, the function approaches the finite value 0.
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20:38:51 query problem 7.8.18 integral of 1 / (`theta^2+1) from 1 to infinity
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RESPONSE --> integral of 1 / (`theta^2+1) from 1 to infinity = integral of 1 / (`theta^2+1^2) from 1 to infinity = (This fits formula V.24.) limit of arctan `theta from 1 to b as b approaches infinity = arctan(9*10^99) - acrtan(1) = 0.13389
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20:39:58 does the integral converge or diverge, and why?
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RESPONSE --> The integral converges. It approaches the finite value 0.13389 as x approaches infinity.
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20:57:27 If you have not already stated it, with what convergent or divergent integral did you compare the given integral?
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RESPONSE --> It could have been compared to the integral of 1/x^p from 1 to infinity for p > 1. The function, `sqrt (`theta^2 + 1), will always be slightly larger than `theta itself. In other words, `theta raised to a power greater than 1.
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19:03:21 query problem 7.8.19 (3d edition #20) convergence of integral from 0 to 1 of 1 / `sqrt(`theta^3 + `theta)
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RESPONSE --> The problem you have there is #20 in the 4th edition. The `sqrt(`theta^3 + `theta) = `theta^p where p is greater than 1. The integral from 0 to 1 of 1 / `sqrt(`theta^3 + `theta) is similar to the integral from 0 to 1 of 1 / `x^p, where p is greater than 1. It follows that the integral diverges.
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19:03:45 11-27-2007 19:03:45 does the integral converge or diverge, and why?
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NOTES ------->
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19:04:03 If you have not already stated it, with what convergent or divergent integral did you compare the given integral?
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RESPONSE -->
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21:07:46 Query problem 8.1.5. Riemann sum and integral inside x^2 + y^2 = 10 within 1st quadrant, using horizontal strip of width `dy.
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RESPONSE --> x^2 + y^2 = 10 s = is the length of each slice area of each slice = s`dy s is the length of the adjacent side of a triangle with hypotenuse `sqrt(10) and height y. s in terms of y is s = `sqrt(10 - y^2) The sum of `sqrt(10 - y^2) 'dy The integral from 0 to `sqrt(10) of `sqrt(10 - y^2) dy = 5`pi / 2 = area of the region
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21:12:09 Give the Riemann sum and the definite integral it approaches.
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RESPONSE --> The limit as n approaches infinity, so `dy approaches 0 of the sum from i = 1 to n of `sqrt(10 - y^2) 'dy The integral from 0 to `sqrt(10) of `sqrt(10 - y^2) dy = 5`pi / 2 = area of the region
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21:12:23 Give the exact value of your integral.
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RESPONSE --> 5`pi / 2
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21:31:13 Query problem 8.1.12. Half disk radius 7 m thickness 10 m, `dy slice parallel to rectangular base.
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RESPONSE --> 1/2 `pi r^2 * L = area of the half cylinder `dy * s * 10 approximates the volume of the slice s is twice the length of the adjacent side of a triangle with hypotenuse 7 and height y. s = 2 * `sqrt(49 - y^2) The limit as n approaches infinity, so `dy approaches 0 of the sum from i = 1 to n of 10 * 2 * `sqrt(49 - y^2)`dy = The integral from 0 to 7 of 20 * `sqrt(49 - y^2) dy = 245 `pi m^3 = volume of the region
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21:31:53 Give the Riemann sum and the definite integral it approaches.
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RESPONSE --> The limit as n approaches infinity, so `dy approaches 0 of the sum from i = 1 to n of 10 * 2 * `sqrt(49 - y^2)`dy = The integral from 0 to 7 of 20 * `sqrt(49 - y^2) dy = 245 `pi m^3 = volume of the region
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21:32:05 Give the exact value of your integral.
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RESPONSE --> 245 `pi m^3
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22:41:01 query problem 8.2.11 arc length x^(3/2) from 0 to 2
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RESPONSE --> f(x) = `sqrt(x^3) = x^(3/2) f'(x) = 3x^(1/2) / 2 arc length of the graph of f(x) from 0 to 2 = the integral from 0 to 2 of `sqrt(1+(3x^(1/2) / 2)^2) dx = approximately 3.53
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22:41:11 what is the arc length?
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RESPONSE --> approximately 3.53
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22:41:22 What integral do you evaluate obtain the arc length?
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RESPONSE --> the integral from 0 to 2 of `sqrt(1+(3x^(1/2) / 2)^2) dx
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22:50:24 What is the approximate arc length of a section corresponding to an increment `dx near coordinate x on the x axis?
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RESPONSE --> sqrt(1+(3x^(1/2) / 2)^2) 'dx
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22:51:53 What is the slope of the graph near the graph point with x coordinate x?
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RESPONSE --> (3x^(1/2) / 2) `dx / `dx
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22:54:53 How is this slope related to the approximate arc length of the section?
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RESPONSE --> (3x^(1/2) / 2) 'dx / `dx sqrt(1+(3x^(1/2) / 2)^2) `dx The slope is the quotient of side opp. and side adj. of a triangle with hypotenuse = approx. arc length.
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23:15:30 query problem 8.2.31 volume of region bounded by y = e^x, x axis, lines x=0 and x=1, cross-sections perpendicular to the x axis are squares
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RESPONSE --> the approx. area of a slice = `dx * y y = 1 - x^2 total volume = the sum of (1 - x^2) `dx The limit as `dx approaches 0 gives the definite integral from 0 to 1 of (1 - x^2) dx the definite integral from 0 to 1 of (1 - x^2) dx = 2/3
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23:15:39 what is the volume of the region?
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RESPONSE --> 2 / 3
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23:16:40 What integral did you evaluate to get the volume?
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RESPONSE --> The definite integral from 0 to 1 of (1 - x^2) dx = 2/3
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23:17:11 What is the cross-sectional area of a slice perpendicular to the x axis at coordinate x?
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RESPONSE --> the approx. area of a slice = `dx * y y = 1 - x^2
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23:17:46 What is the approximate volume of a thin slice of width `dx at coordinate x?
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RESPONSE --> total volume = the sum of (1 - x^2) `dx
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23:18:01 How the you obtain the integral from the expression for the volume of the thin slice?
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RESPONSE --> the approx. area of a slice = `dx * y y = 1 - x^2 total volume = the sum of (1 - x^2) `dx The limit as `dx approaches 0 gives the definite integral from 0 to 1 of (1 - x^2) dx the definite integral from 0 to 1 of (1 - x^2) dx = 2/3
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23:18:13 Query Add comments on any surprises or insights you experienced as a result of this assignment.
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RESPONSE --> fun
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