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How do you find the volume of the solid?

How do you find the volume of the solid?

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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How do you find the volume of the solid?

3 Using the curves in Example 4, find the volume of the solid if the curves are rotated about the y-axis instead of the x-axis. HINT: You will need to integrate in y, not x! Do you get the same volume as in Example 4? Why or why not? 4 A common mistake in setting up the integral in Example 4 is to write the integral as √(x-x²) dx instead of [r((x)³ −(x²)³)dx. dx. The integral [7(x-x²)² dx actually gives a 0 volume of a different solid using a disk instead of a washer. On the axes to the right, draw and shade the region of area used to get the volume [(x-x²)² dx. 0 EXTRA CREDIT: Can you roughly sketch the solid too?
Transcribed Image Text:3 Using the curves in Example 4, find the volume of the solid if the curves are rotated about the y-axis instead of the x-axis. HINT: You will need to integrate in y, not x! Do you get the same volume as in Example 4? Why or why not? 4 A common mistake in setting up the integral in Example 4 is to write the integral as √(x-x²) dx instead of [r((x)³ −(x²)³)dx. dx. The integral [7(x-x²)² dx actually gives a 0 volume of a different solid using a disk instead of a washer. On the axes to the right, draw and shade the region of area used to get the volume [(x-x²)² dx. 0 EXTRA CREDIT: Can you roughly sketch the solid too?
Example 4 Using the Washer Method The region R enclosed by the curves y = x and y = x² is rotated about the x-axis. Find the volume of the resulting solid. Solution The curves y = x and y = x² intersect at the points (0, 0) and (1, 1). The region between them, the solid of rotation, and a cross-section perpendicular to the x-axis are shown in Figure 8. A cross-section in the plane P, has the shape of a washer (an annular ring) with inner radius ² and outer radius x, so we find the cross-sectional area by subtracting the area of the inner circle from the area of the outer circle: Therefore we have A(x) = πx² - (x²)² = π(x²-x²) v = [¹ A(z) dr = [¹ (2² - r¹) dr dx
Transcribed Image Text:Example 4 Using the Washer Method The region R enclosed by the curves y = x and y = x² is rotated about the x-axis. Find the volume of the resulting solid. Solution The curves y = x and y = x² intersect at the points (0, 0) and (1, 1). The region between them, the solid of rotation, and a cross-section perpendicular to the x-axis are shown in Figure 8. A cross-section in the plane P, has the shape of a washer (an annular ring) with inner radius ² and outer radius x, so we find the cross-sectional area by subtracting the area of the inner circle from the area of the outer circle: Therefore we have A(x) = πx² - (x²)² = π(x²-x²) v = [¹ A(z) dr = [¹ (2² - r¹) dr dx
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