Solutions manual for mechanics and thermodynamics
.pdf
111
R R
8.Evaluate x2dx and 3x3dx.
SOLUTION
y = R f dx with f(x) ¥ dxdy A) the derivative function is f(x) = x2 = dxdy . Thus the original function must be 13 x3 + c. Thus
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x2dx = 3x3 + c |
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B) the derivative function is f(x) |
= 3x3 = |
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. Thus the original |
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41 x4 + c¥. Thus |
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dx |
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function must be 3 |
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3x3dx = 4x4 + 3c |
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or = |
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x4 + c0 |
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where I have written c0 ¥ 3c.
112 |
CHAPTER 17. REVIEW OF CALCULUS |
9.What is the area under the curve f(x) = x between x1 = 0 and x2 = 3? Work out your answer i) graphically and ii) with the integral.
SOLUTION f(x) = x
The area of the triangle between x1 = 0 and x1 = 3 is 12 £ Base £ Height = 12 £ 3 £ 3 = 4:5
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x dx = |
2x2 |
+ c 0 |
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µ232 + c ° |
µ202 |
+ c |
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µ |
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° c |
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9 |
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= 4:5 |
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in agreement with the graphical method.