| Calculus II Assignment # 1 |
This assignment covers
The Mean Value Theorem for Integrals
Integration by Substitution (change of variable)
Area Under a Curve
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Questions
1) Given f (x) = x² + 2x - 5, find z on the interval [1, 4] such that
z satisfies the conclusion of the Mean Value Theorem for Integrals.
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2) Use Substitution or Change or Variable to Evaluate These Integrals
| a) |
b) |
| c) |
d) |
| e) |
f) |
| g) |
h) |
| i) |
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3) 
find the area of the region under the graph of f from 0 to 2.
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Solutions
1)
We need f(z) (b - a) = 21 so 3(z² + 2z - 5) = 21
Using the quadratic formula, we get 
since z must be on (1, 4).
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2) Use Substitution or Change or Variable to Evaluate These Integrals
| a) take the cube root to simplify it.
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b) Let u = 5x + 1, so du/5 = dx
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| c) Let u = 1+ z ³, so du/3 = z² dz also, when z = 0, u = 1 and z = 1, u = 2
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d) simplify the integrand to
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| e) Let u = w² + 2w , so du/2 = (w + 1)dw also, when w = 1, u = 3 and w = 2, u = 8
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f) If we let u = 2x ³, then du/6 = x² dx
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| g) Let u = sin 3x, so du/3 = cos 3x dx
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h) Let u = 3 + 5 sin x, so du/5 = cos x dx when x = 0, u = 3 and x = o /2, u = 8
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| i) Let u = cos 2x, so du/-2 = sin 2x dx when x = 0, u = 1 and x = o /4, u = 0
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3) Let u = x 5 + 4, so du / 5 = x 4 dx
Also, when x = 0, u = 4 and when x = 2, u = 36
Area under the curve is
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