Chapter 4: Q. 7 (page 404)

Riemann sums: Calculate each of the following Riemann sum
approximations for the definite integral of f on [a, b], using the
given value of n.
The left sum for $f (x) = \sqrt{x}$ on $[0, 4]$ , $n = 4$ .

Short Answer

Expert verified

The value is $6 \cdot 146$ .

Step by step solution

Step 1. Given information .

Consider the given function $f (x) = \sqrt{x}$ .

Step 2. Formula used .

The Riemann sum formula for left sum is $L_{n} = \sum_{i = 0}^{n - 1} f (x_{i}) \cdot δ x$ where $L_{n}$ is the left sum with the n rectangle and $δ x = \frac{b - a}{n}$ .

Step 3. Find the value .

Divided the open interval $[0, 4]$ in 4 sub intervals $(0, 1), (1, 2), (2, 3), (3, 4)$ where $x_{0} = 0, x_{1} = 1, x_{2} = 2, x_{3} = 3, x_{4} = 4$

$δ x = \frac{b - a}{h} = \frac{4 - 0}{4} = 1$

Substitute the value of $δ x$ and $x_{0} . . . . . . . . x_{4}$ in the given formula .

$L_{n} = \sum_{i = 0}^{4} f (x_{i}) \cdot δ x$

$L_{n} = f (x_{0}) \cdot δ x + f (x_{1}) \cdot δ x + f (x_{2}) \cdot δ x + f (x_{3}) \cdot δ x_{3} + f (x_{4}) \cdot δ x = 0 \times 1 + 1 \times 1 + \sqrt{2} \times 1 + \sqrt{3} \times 1 + 2 \times 1 = 3 + \sqrt{2} + \sqrt{3} = 6 \cdot 146$

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Riemann sums: Calculate each of the following Riemann sum
approximations for the definite integral of f on [a, b], using the
given value of n.
The left sum for $f (x) = \sqrt{x}$ on $[0, 4]$ , $n = 4$ .

Short Answer

Step by step solution

Step 1. Given information .

Step 2. Formula used .

Step 3. Find the value .

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Riemann sums: Calculate each of the following Riemann sumapproximations for the definite integral of f on [a, b], using thegiven value of n.The left sum for fx=xon 0,4 ,n=4.

Short Answer

Step by step solution

Step 1. Given information .

Step 2. Formula used .

Step 3. Find the value .

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Decision Maths

Applied Mathematics

Mechanics Maths

Geometry

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

Riemann sums: Calculate each of the following Riemann sum
approximations for the definite integral of f on [a, b], using the
given value of n.
The left sum for $f (x) = \sqrt{x}$ on $[0, 4]$ , $n = 4$ .