Chapter 6: Q. 45 (page 511)

Consider the region between the graphs of $f (x) = 5 - x$ and $g (x) = 2 x$ on . For each line of rotation given in Exercises 45 and 46, use definite integrals to find the volume of the resulting solid.

Short Answer

Expert verified

The volume of the solid is $\frac{1925}{27} π$

Step by step solution

Step 1. Given Information

The given figure is

$f (x) = g (x) 5 - x = 2 x x = \frac{5}{3}$

Step 2. Finding Volume

$V_{1} = π \int_{1}^{\frac{5}{3}} {(5 - x)}^{2} - {(2 x)}^{2} dx V_{1} = π \int_{1}^{\frac{5}{3}} (25 + x^{2} - 10 x - 4 x^{2}) dx V_{1} = π \int_{1}^{\frac{5}{3}} (- 3 x^{2} - 10 x + 25) dx V_{1} = π {[- x^{3} - 5 x^{2} + 25 x]}_{1}^{\frac{5}{3}} V_{1} = π [(- \frac{125}{27} - \frac{125 \times 3}{9 \times 3} + \frac{125 \times 9}{3 \times 9}) - (- 1 - 5 + 25)] V_{1} = π [\frac{625}{27} - 19] = 4.14 π$

and

$V_{2} = π \int_{\frac{5}{3}}^{4} {(2 x)}^{2} - {(5 - x)}^{2} dx V_{2} = π \int_{\frac{5}{3}}^{4} (4 x^{2} - 25 - x^{2} + 10 x) dx V_{2} = π \int_{\frac{5}{3}}^{4} (3 x^{2} + 10 x - 25) dx V_{2} = π {[x^{3} + 5 x^{2} - 25 x]}_{\frac{5}{3}}^{4} V_{2} = π [(64 + 80 - 100) - (\frac{125}{27} + \frac{125 \times 3}{9 \times 3} - \frac{125 \times 9}{3 \times 9})] = π [44 + \frac{625}{27}] = 67.14 π$

Step 3. Finding Final Volume

$V = V_{1} + V_{2} V = 4.14 π + 67.14 π = 71.29 π$

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Consider the region between the graphs of $f (x) = 5 - x$ and $g (x) = 2 x$ on . For each line of rotation given in Exercises 45 and 46, use definite integrals to find the volume of the resulting solid.

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Finding Volume

Step 3. Finding Final Volume

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Consider the region between the graphs of fx=5-xand gx=2xon . For each line of rotation given in Exercises 45 and 46, use definite integrals to find the volume of the resulting solid.

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Finding Volume

Step 3. Finding Final Volume

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Statistics

Pure Maths

Applied Mathematics

Decision Maths

Logic and Functions

Study anywhere. Anytime. Across all devices.

Consider the region between the graphs of $f (x) = 5 - x$ and $g (x) = 2 x$ on . For each line of rotation given in Exercises 45 and 46, use definite integrals to find the volume of the resulting solid.