Give precise mathematical definitions or descriptions of each of the concepts that follow. Then illustrate the definition with a graph or algebraic example, if possible Q. 2 The volume of a solid with... [FREE SOLUTION]

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Chapter 6
Give precise mathematical definitions or descriptions of each of the concepts that follow. Then illustrate the definition with a graph or algebraic example, if possible

Chapter 6: Give precise mathematical definitions or descriptions of each of the concepts that follow. Then illustrate the definition with a graph or algebraic example, if possible (page 573)

Q. 2 The volume of a solid with disk cross-sections whose radii are given by $r (x)$ on $[a, b]$ , both as a limit of Riemann sums and as a definite integral.

Short Answer

Expert verified

The volume by using definite integral,

$V = 蟺 {鈭�}_{a}^{b} {[r (x)]}^{2} d x$

The volume as a limit of Riemann sums is,

$蟺 {鈭�}_{a}^{b} {[r (x)]}^{2} d x = 蟺 \lim_{n 鈫� 鈭�} {鈭�}_{i = 1}^{鈭�} {[r (x_{i})]}^{2} 鈭� x Where, 鈭� x = \frac{b - a}{n}, 鈭� x_{i} = a + 鈭� x 路 i$

Step by step solution

Volume of a solid by using definite integral.

If the radius is $r (x)$

Then, the volume of the solid on the interval $[a, b]$ is given by a definite integral,

role="math" localid="1662975511626" $V = 蟺 {鈭�}_{a}^{b} {[r (x)]}^{2} d x$

Volume of solid by using the limit of Riemann sums.

The definite integral of a continuous function $r (x)$ over the interval $[a, b]$ denoted by $V = 蟺 {鈭�}_{a}^{b} {[r (x)]}^{2} d x$ , is the limit of a Riemann sum as the number of subdivisions approaches infinity.

Therefore,

$蟺 {鈭�}_{a}^{b} {[r (x)]}^{2} d x = 蟺 \lim_{n 鈫� 鈭�} {鈭�}_{i = 1}^{鈭�} {[r (x_{i})]}^{2} 鈭� x Where, 鈭� x = \frac{b - a}{n}, 鈭� x_{i} = a + 鈭� x 路 i$

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Q. 2 The volume of a solid with disk cross-sections whose radii are given by $r (x)$ on $[a, b]$ , both as a limit of Riemann sums and as a definite integral.

Short Answer

Step by step solution

Volume of a solid by using definite integral.

Volume of solid by using the limit of Riemann sums.

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91影视

Q. 2 The volume of a solid with disk cross-sections whose radii are given by r(x)on a,b, both as a limit of Riemann sums and as a definite integral.

Short Answer

Step by step solution

Volume of a solid by using definite integral.

Volume of solid by using the limit of Riemann sums.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Statistics

Decision Maths

Discrete Mathematics

Applied Mathematics

Pure Maths

Study anywhere. Anytime. Across all devices.

Q. 2 The volume of a solid with disk cross-sections whose radii are given by $r (x)$ on $[a, b]$ , both as a limit of Riemann sums and as a definite integral.