Chapter 6: Q. 43 (page 511)

Consider the region between the graph of $f (x) = 4 - x^{2}$ and the line $y = 5$ on For each line of rotation given in Exercises 41–44, use definite integrals to find the volume of the resulting solid.

Short Answer

Expert verified

The volume of the solid is $16 π$

Step by step solution

Step 1. Given Information

The given figure is

$f (x) = 4 - x^{2} = y x = \sqrt{4 - y} = g (y)$

Step 2. Finding Volume

$V_{1} = π \int_{0}^{4} {(3 - (\sqrt{4 - y}) - 1)}^{2} dy V_{1} = π \int_{0}^{4} (12 - y - 6 \sqrt{4 - y}) dy V_{1} = π {[12 y - \frac{y^{2}}{2} - 6 (\frac{{(4 - y)}^{\frac{3}{2}}}{- \frac{3}{2}})]}_{0}^{4} = V_{1} = π {[12 y - \frac{y^{2}}{2} + 4 {(4 - y)}^{\frac{3}{2}}]}_{0}^{4} V_{1} = π [48 - 8 + 4 (0) - (32)] = 8 π$

and

$V_{2} = π \int_{4}^{5} (9 - 1) d y = 8 π \int_{4}^{5} d y V_{2} = 8 π [5 - 4] = 8 π$

Now,

$V = V_{1} + V_{2} V = 8 π + 8 π = 16 π$

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Consider the region between the graph of $f (x) = 4 - x^{2}$ and the line $y = 5$ on For each line of rotation given in Exercises 41–44, 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

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Consider the region between the graph of fx=4-x2and the liney=5on For each line of rotation given in Exercises 41–44, 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

One App. One Place for Learning.

Most popular questions from this chapter

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Consider the region between the graph of $f (x) = 4 - x^{2}$ and the line $y = 5$ on For each line of rotation given in Exercises 41–44, use definite integrals to find the volume of the resulting solid.