Rectangle Approximations for the Definite Integrals Worksheet

**Page 1**

1.

Find the approximate value of ${\int}_{1}^{3}$$x$^{3} $\mathrm{dx}$ by using 4 rectangles of equal width.

a. | 25.8125 | ||

b. | 55 | ||

c. | 24.125 | ||

d. | 27.5 | ||

e. | 20.00 |

The height of each rectangle is determined by the value of

The table represents the values of

1 | 1.5 | 2 | 2.5 | 3 | |

1 | 3.375 | 8 | 15.625 | 27 |

So, the approximate value of

= (0.5)

= (0.5)(1) + (0.5)(3.375) + (0.5)(8) + (0.5)(15.625) + (0.5)(27)

= 27.5

Correct answer : (4)

2.

Find the approximate value of ${\int}_{1}^{3}$ $e$^{$x$} $\mathrm{dx}$ by using 4 rectangles of equal width.

a. | 17.366 | ||

b. | 23.398 | ||

c. | 44.077 | ||

d. | 46.796 | ||

e. | 22.036 |

The corresponding height of each rectangle is determined by the value of

The table represents the values of

1 | 1.5 | 2 | 2.5 | 3 | |

2.7183 | 4.481 | 7.389 | 12.123 | 20.085 |

So, the approximate value of

= (0.5)

= (0.5)(2.7183) + (0.5)(4.481) + (0.5)(7.389) + (0.5)(12.123) + (0.5)(20.085)

= 23.398

Correct answer : (2)

3.

Find the approximate value of ${\int}_{1}^{4}$ln ($x$ + 2) $\mathrm{dx}$ using 6 rectangles of equal width.

a. | 9.824 | ||

b. | 5.172 | ||

c. | 10.344 | ||

d. | 8.215 | ||

e. | 6.963 |

The corresponding height of each rectangle is determined by the value of

The table represents the values of

1 | 1.5 | 2 | 2.5 | 3 | 3.5 | 4 | |

1.098 | 1.252 | 1.386 | 1.504 | 1.609 | 1.704 | 1.791 |

So, the approximate value of

= (0.5)

= (0.5)(1.098) + (0.5)(1.252) + (0.5)(1.386) + (0.5)(1.504) + (0.5)(1.609) + (0.5)(1.704) + (0.5)(1.791)

= 5.172

Correct answer : (2)

4.

Find the approximate value of ${\int}_{1}^{3}$ $\frac{1}{x}$ using 4 rectangles of equal width.

a. | 1.098 | ||

b. | 1.25 | ||

c. | 1.0835 | ||

d. | 2.90 | ||

e. | 1.45 |

The corresponding height of each rectangle is determined by the value of

The table represents the value of

1 | 1.5 | 2 | 2.5 | 3 | |

1 | 0.667 | 0.5 | 0.4 | 0.333 |

So, the approximate value of

= (0.5)(1) + (0.5)(0.667) + (0.5)(0.5) + (0.5)(0.4) + (0.5)(0.333)

= 1.45

Correct answer : (5)

5.

Find the approximate value of ${\int}_{1}^{4}$($x$^{2} - 3$x$ + 6) $\mathrm{dx}$ using 6 rectangles of equal width.

a. | 20.525 | ||

b. | 20.125 | ||

c. | 37.25 | ||

d. | 18.625 | ||

e. | 16.50 |

The corresponding height of each rectangle is determined by the value of

The table represents the values of

1 | 1.5 | 2 | 2.5 | 3 | 3.5 | 4 | |

4 | 3.75 | 4 | 4.75 | 6 | 7.75 | 10 |

The approximate value of

= (0.5)(4) + (0.5)(3.75) + (0.5)(4)+ (0.5)(4.75) + (0.5)(6) + (0.5)(7.75) + (0.5)(10)

= 20.125

Correct answer : (2)

6.

Find the approximate value of ${\int}_{-2}^{2}$ ln($x$^{2} + 2) $\mathrm{dx}$ by using 4 rectangles of equal width.

a. | 5.373 | ||

b. | 3.582 | ||

c. | 6.471 | ||

d. | 5.778 | ||

e. | 4.680 |

The corresponding height of each rectangle is determined by the value of

The table represents the values of

- 2 | - 1 | 0 | 1 | 2 | |

1.791 | 1.098 | 0.693 | 1.098 | 1.791 |

So, the approximate value of

= (1)(1.791) + (1)(1.098) + (1)(0.693) + (1)(1.098) + (1)(1.791)

= 6.471

Correct answer : (3)

7.

Find the approximate value of ${\int}_{-3}^{-1}$ $e$^{(2$x$ + 4)} $\mathrm{dx}$ by using 4 rectangles of equal width.

a. | 5.894 | ||

b. | 8.077 | ||

c. | 3.141 | ||

d. | 6.394 | ||

e. | 11.7893 |

The corresponding height of each rectangle is determined by the value of

The table represents the values of

- 3 | - 2.5 | - 2 | - 1.5 | - 1 | |

0.135 | 0.367 | 1 | 2.7183 | 7.389 |

So, the approximate value of

= (0.5)(0.135) + (0.5)(0.367) + (0.5)(1) + (0.5)(2.7183) + (0.5)(7.389)

= 5.894

Correct answer : (1)