Chapter 4: Q. 15 (page 404)

Calculating definite integrals with limits of Riemann sums: Calculate
the exact value of each the following definite integrals by
setting up a general Riemann sum and then taking the limit
as n→∞.
$\int_{- 2}^{3} (4 - x^{2}) d x$

Short Answer

Expert verified

The definite integral is $\frac{25}{3}$ .

Step by step solution

Step 1. Given information .

Consider the given integral $\int_{- 2}^{3} (4 - x^{2}) d x$ .

Step 2. Formula used .

$\int_{a}^{b} f (x) d x = \lim_{n \to \infty} \sum_{k = 1}^{n} f (a + k \cdot δ x) \cdot δ x$

Step 3. Find the integral.

$δ x = \frac{3 + 2}{n} = \frac{5}{n} a + k \cdot δ x = - 2 + k \frac{5}{n} f (a + k \cdot δ x) \cdot δ x = [4 - {(\frac{5 k}{n} - 2)}^{2}] \cdot \frac{5}{n} = \frac{100 k}{n^{2}} - \frac{125 k^{2}}{n^{3}} \int_{a}^{b} f (x) d x = \lim_{n \to \infty} \sum_{k = 1}^{n} f (a + k \cdot δ x) \cdot δ x = \lim_{n \to \infty} \frac{100}{n^{2}} \sum_{k = 1}^{n} k - \frac{125}{n^{3}} \sum_{k = 1}^{n} k^{2} = \lim_{n \to \infty} \frac{100}{n^{2}} \cdot \frac{n (n + 1)}{2} - \frac{125}{n^{3}} \cdot \frac{n (n + 1) (2 n + 1)}{6} = \lim_{n \to \infty} \frac{100}{n^{2}} \cdot \frac{n^{2} (1 + \frac{1}{n})}{2} - \frac{125}{n^{3}} \cdot \frac{n^{3} (1 + \frac{1}{n}) (2 + \frac{1}{n})}{6} = \frac{25}{3}$

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Calculating definite integrals with limits of Riemann sums: Calculate
the exact value of each the following definite integrals by
setting up a general Riemann sum and then taking the limit
as n→∞.
$\int_{- 2}^{3} (4 - x^{2}) d x$

Short Answer

Step by step solution

Step 1. Given information .

Step 2. Formula used .

Step 3. Find the integral.

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Calculating definite integrals with limits of Riemann sums: Calculatethe exact value of each the following definite integrals bysetting up a general Riemann sum and then taking the limitas n→∞.∫-234-x2dx

Short Answer

Step by step solution

Step 1. Given information .

Step 2. Formula used .

Step 3. Find the integral.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Pure Maths

Discrete Mathematics

Mechanics Maths

Theoretical and Mathematical Physics

Geometry

Study anywhere. Anytime. Across all devices.

Calculating definite integrals with limits of Riemann sums: Calculate
the exact value of each the following definite integrals by
setting up a general Riemann sum and then taking the limit
as n→∞.
$\int_{- 2}^{3} (4 - x^{2}) d x$