Calculus II Assignment # 1 |

This assignment covers

**The Mean Value Theorem for Integrals**

**Integration by Substitution (change of variable)**

**Area Under a Curve**

.

**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.

.

2) Use Substitution or Change or Variable to Evaluate These Integrals

a) | b) |

c) | d) |

e) | f) |

g) | h) |

i) |

.

3)

find the area of the region under the graph of *f* from 0 to 2.

.

**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).

.

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 dzalso, 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)dwalso, 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 |

i) Let u = cos 2x, so du/-2 = sin 2x dxwhen |

.

3) Let *u = x ^{ 5} + 4*, so

Also, when

Area under the curve is

.

*(Back to Caculus II MathRoom Index)*

.

*(all content of the MathRoom Lessons **© Tammy the Tutor; 2002 - 2005).*