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Basic Definite Integration, area under the curve

  • Thread starter FaraDazed
  • Start date
  • #1
347
2

Homework Statement


Evaluate:

A:
[tex]
\int^{16}_0 (\sqrt{x} - 1)\,dx
[/tex]

B:
[tex]
\int^4_1 \frac{2}{\sqrt{x}}\,dx
[/tex]

C:
[tex]
\int^6_2 \frac{4}{x^3}\,dx
[/tex]


Homework Equations



n/a

The Attempt at a Solution


Its part C which I think I have done wrong, but the others could be too.

Part A:
[tex]
\int^{16}_0 (\sqrt{x} - 1)\,dx = \int^{16}_0 (x^{\frac{1}{2}}-1)\,dx = [\frac{2}{3}x^{1.5}-1x]^{16}_0 = [112]-[0] = 112 \,sq\, units
[/tex]

Part B:
[tex]
\int^4_1 \frac{2}{\sqrt{x}}\,dx = \int^4_1 2x^{-\frac{1}{2}}\,dx = [4x^{\frac{1}{2}}]^4_1 = [8] - [4] = 4 \,sq\, units
[/tex]

C:
[tex]
\int^6_2 \frac{4}{x^3}\,dx = \int^6_2 4x^{-3}\,dx = [-2x^{-2}]^6_2 = [-\frac{1}{18}] - [-\frac{1}{2}] = 0.44 \,sq\, units
[/tex]
 
Last edited:

Answers and Replies

  • #2
216
0

Homework Statement


Evaluate:

A:
[tex]
\int^{16}_0 (\sqrt{x} - 1)\,dx
[/tex]

B:
[tex]
\int^4_1 \frac{2}{\sqrt{x}}\,dx
[/tex]

C:
[tex]
\int^6_2 \frac{4}{x^3}\,dx
[/tex]


Homework Equations



n/a

The Attempt at a Solution


Its part C which I think I have done wrong, but the others could be too.

Part A:
[tex]
\int^{16}_0 (\sqrt{x} - 1)\,dx = \int^{16}_0 (x^{\frac{1}{2}}-1)\,dx = [\frac{2}{3}x^{1.5}-1x]^{16}_0 = [112]-[0] = 112 sq units
[/tex]

Part B:
[tex]
\int^4_1 \frac{2}{\sqrt{x}}\,dx = \int^4_1 2x^{-\frac{1}{2}}\,dx = [4x^{\frac{1}{2}}]^4_1 = [8] - [4] = 4 sq units
[/tex]

C:
[tex]
\int^6_2 \frac{4}{x^3}\,dx = \int^6_2 4x^{-3}\,dx = [-2x^{-2}]^6_2 = [-\frac{1}{18}] - [-\frac{1}{2}] = 0.44 sq units
[/tex]
first is incorrect - others are fine (assuming you mean 4/9 for the third).

apply the limits again
 
  • #3
347
2
first is incorrect - others are fine (assuming you mean 4/9 for the third).

apply the limits again
Ah yeah, dont know what i was thinking there!

Should it be...
[tex]
[\frac{80}{3}]-[0] = \frac{80}{3} sq\,units
[/tex]

And yeah 4/9 is what I meant its just the way it came out on my calc I couldnt work out what fraction produced that recurring decimal :)

Thanks for your help.
 

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