Chapter 4: Q. 14 (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→∞.
$\int_{0}^{4} x^{2} d x$

Short Answer

Expert verified

The definite value is $\frac{64}{3}$ .

Step by step solution

Step 1. Given information .

Consider the given definite integral $\int_{0}^{4} x^{2} d x$ .

Step 2. Formula used .

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

Step 3. Calculating the integral .

$δ x = \frac{4 - 0}{n} = \frac{4}{n} f (a + k \cdot δ x) = {(\frac{4 k}{n})}^{2} \cdot \frac{4}{n} = \frac{64 k^{2}}{n^{3}} \int_{a}^{b} f (x) d x = \lim_{n \to \infty} \sum_{k = 1}^{n} f (a + k \cdot δ x) δ x = \lim_{n \to \infty} \sum_{k = 1}^{n} \frac{64}{n^{3}} \cdot k^{2} = \lim_{n \to \infty} \frac{64}{n^{3}} \cdot \frac{n (n + 1) (2 n + 1)}{6} = \lim_{n \to \infty} \frac{64}{n^{3}} \cdot \frac{n^{3} (1 + \frac{1}{n}) (2 + \frac{1}{n})}{6} = \frac{64}{3}$

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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→∞.
$\int_{0}^{4} x^{2} d x$

Short Answer

Step by step solution

Step 1. Given information .

Step 2. Formula used .

Step 3. Calculating 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→∞.∫04x2dx

Short Answer

Step by step solution

Step 1. Given information .

Step 2. Formula used .

Step 3. Calculating the integral .

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

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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→∞.
$\int_{0}^{4} x^{2} d x$