Chapter 6: Q. 60 (page 524)

The area centered on the line $x = - 1$ between the graph of $f (x) = x^{2} + 2$ and the x-axis on $[1, 3] .$

Short Answer

Expert verified

The volume of the solid of revolution is $\frac{248}{3} π .$

Step by step solution

Given information

Consider the given function,

$f (x) = x^{2} + 2$ $f (x) = x^{2} + 2$

Calculation.

The region is enclosed by the expression $f (x) = x^{2} + 2$ and the x-axis on $[- 1, 3] .$

Use the shell method with shells on the x-axis having radius $x - (- 1) = x + 1$ , where x ranges from localid="1661340316266" $- 1 to 3$ , and height $f (x) = x^{2} + 2$ , as the region is revolved around the line $x = - 1 .$

Thus, the volume is provided by

localid="1661340332870" $\begin{array}{rcl} Volume & = & 2 π \int_{- 1}^{3} (x + 1) (x^{2} + 2) d x \\ = & 2 π \int_{- 1}^{3} (x^{3} + x^{2} + 2 x + 1) d x \\ = & 2 π {[\frac{x^{4}}{4} + \frac{x^{3}}{3} + x^{2} + x]}_{- 1}^{3} \\ = & 2 π [(\frac{81}{4} + 9 + 9 + 3) - (\frac{1}{4} - \frac{1}{3} + 1 - 1)] \end{array}$

Therefore the volume of the solid of revolution is localid="1661340343180" $\frac{248}{3} π$ .

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The area centered on the line $x = - 1$ between the graph of $f (x) = x^{2} + 2$ and the x-axis on $[1, 3] .$

Short Answer

Step by step solution

Given information

Calculation.

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The area centered on the line x=-1between the graph of f(x)=x2+2and the x-axis on [1,3].

Short Answer

Step by step solution

Given information

Calculation.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Applied Mathematics

Decision Maths

Geometry

Statistics

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

The area centered on the line $x = - 1$ between the graph of $f (x) = x^{2} + 2$ and the x-axis on $[1, 3] .$