- #1
Timiop2008
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Please Help! - Integration, Rate of Change, and Volume of Revolution Questions
Hi
I have completes these following questions but am not sure if I have done them correctly as it is a long time since i studied these topics. I would really appreciate any help. :-)
1 Simplify the following as far as Possible
(a) ∫4e2x dx (2 marks)
(b) ∫(4/x) dx between 1 and 3 (3 marks)
(c) ∫(3x+4)5 dx (2 marks)
(d) ∫((x/2)+1)2 dx (2 marks)
2 Find the area bound by the graph y=3/x, the lines y=1, y=3, and the y-axis (3 marks)
3 Find the Volume of the solid formed by the line y=ex+1 between 0≤x≤1 when it is rotated through 360 degrees about the x axis. Leave your answer in terms of e. ( 5 marks)
4
(a) The radius of a sphere is increasing at a rate of 0.8mm/s. Find the rate at which the volume of the sphere is increasing when r=10mm. Leave your answer in terms of pi (3 marks)
(b) The Volume of the same sphere then starts to decrease at a rate of 27m3/s. Find the rate at which the radius is decreasing when the volume is 3m3
Leave your answer in terms of Pi. (5 marks)
My attempts at Solutions:
1
a) ∫4e2x dx
=(4/2)e2 X x2
= 2e2x2 +K
b) (4/x) = 4x-1
4x-1 dx
[4lnx] between 3 and 1
4ln(3) - 4ln(1)
4ln(3)
=4.39
c) ∫(3x+4)5 dx
= ∫243x5+1620x4+4320x3+5760x2+3072x
= (81/2)x6+324x5+1080x4+1920x3+1920x2+1024x +K
d) ∫((x/2)+1)2 dx
= ∫(x2/4)+x+1 dx
= (x3+6x2+12x)/12
= (x/12)3+(1/2x2)+x +K
2) ∫(between 1 and 3) of (3/x) dy
=∫(between 1 and 3) of 3x-1 dy
=[3x-1y] for 1 and 3
= (3x-1X3) - (3x-1)
= 9/x - 3/x
= 6/x +K
3) y=ex+1 0≤x≤1
=∫(between 0 and 1) of pi y2 dx
=∫(between 0 and 1) of pi (ex+1)2 dx
=∫(between 0 and 1) of (e2x+2ex+1) dx
= Pi[1/2e2x+e2x+x] between 0 and 1
= Pi[1/2e2+e2+1] - [1/2e0+e0+0]
= Pi[1/2e2+e2+1] - [1+1+0]
= Pi[1/2e2+e2+1] - 2
= 1/2e2Pi+e2Pi+Pi-2
= 3/2e2Pi+Pi-2
(I do not really know how to attempt the final rates of change question)
Thank You Everyone
Hi
I have completes these following questions but am not sure if I have done them correctly as it is a long time since i studied these topics. I would really appreciate any help. :-)
1 Simplify the following as far as Possible
(a) ∫4e2x dx (2 marks)
(b) ∫(4/x) dx between 1 and 3 (3 marks)
(c) ∫(3x+4)5 dx (2 marks)
(d) ∫((x/2)+1)2 dx (2 marks)
2 Find the area bound by the graph y=3/x, the lines y=1, y=3, and the y-axis (3 marks)
3 Find the Volume of the solid formed by the line y=ex+1 between 0≤x≤1 when it is rotated through 360 degrees about the x axis. Leave your answer in terms of e. ( 5 marks)
4
(a) The radius of a sphere is increasing at a rate of 0.8mm/s. Find the rate at which the volume of the sphere is increasing when r=10mm. Leave your answer in terms of pi (3 marks)
(b) The Volume of the same sphere then starts to decrease at a rate of 27m3/s. Find the rate at which the radius is decreasing when the volume is 3m3
Leave your answer in terms of Pi. (5 marks)
My attempts at Solutions:
1
a) ∫4e2x dx
=(4/2)e2 X x2
= 2e2x2 +K
b) (4/x) = 4x-1
4x-1 dx
[4lnx] between 3 and 1
4ln(3) - 4ln(1)
4ln(3)
=4.39
c) ∫(3x+4)5 dx
= ∫243x5+1620x4+4320x3+5760x2+3072x
= (81/2)x6+324x5+1080x4+1920x3+1920x2+1024x +K
d) ∫((x/2)+1)2 dx
= ∫(x2/4)+x+1 dx
= (x3+6x2+12x)/12
= (x/12)3+(1/2x2)+x +K
2) ∫(between 1 and 3) of (3/x) dy
=∫(between 1 and 3) of 3x-1 dy
=[3x-1y] for 1 and 3
= (3x-1X3) - (3x-1)
= 9/x - 3/x
= 6/x +K
3) y=ex+1 0≤x≤1
=∫(between 0 and 1) of pi y2 dx
=∫(between 0 and 1) of pi (ex+1)2 dx
=∫(between 0 and 1) of (e2x+2ex+1) dx
= Pi[1/2e2x+e2x+x] between 0 and 1
= Pi[1/2e2+e2+1] - [1/2e0+e0+0]
= Pi[1/2e2+e2+1] - [1+1+0]
= Pi[1/2e2+e2+1] - 2
= 1/2e2Pi+e2Pi+Pi-2
= 3/2e2Pi+Pi-2
(I do not really know how to attempt the final rates of change question)
Thank You Everyone