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. b) Let u = 5x + 1, so du/5 = dx c) Let u = 1+ z ³, so du/3 = z² dz also, when z = 0, u = 1 and z = 1, u = 2 d) simplify the integrand to e) Let u = w² + 2w , so du/2 = (w + 1)dw also, when w = 1, u = 3 and w = 2, u = 8 f) If we let u = 2x ³, then du/6 = x² dx g) Let u = sin 3x, so du/3 = cos 3x dx h) Let u = 3 + 5 sin x, so du/5 = cos x dxwhen x = 0, u = 3 and x = o /2, u = 8 i) Let u = cos 2x, so du/-2 = sin 2x dxwhen 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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