Find the volume of the solid obtained by rotating the region bounded by y= x^3, y = 4x, x \geq 0...
Question:
Find the volume of the solid obtained by rotating the region bounded by {eq}y= x^3 {/eq}, {eq}y = 4x {/eq}, {eq}x \geq 0 {/eq} about the {eq}x {/eq}-axis.
Finding the Volume by Washer Method:
According to the washer method, the difference between the two curves results in the volume of the solid that formed when the curves rotate around the axis. Recall that the area of the circle (or disc) is pie times squared radius. Let {eq}f(x) {/eq} and {eq}g(x) {/eq} are two curves. The volume is given by:
$$V = \pi \int_{a}^{b} \left[ \left( f \left(x \right) \right)^2 - \left( g \left(x \right) \right)^2 \right] \ dx $$
Answer and Explanation:
Become a Study.com member to unlock this answer! Create your account
View this answerConsider the regions into:
- {eq}f \left(x \right) = y = 4x {/eq}
- {eq}g \left(x \right) = y = x^3 {/eq}
To equate the two functions to find the...
See full answer below.
Ask a question
Our experts can answer your tough homework and study questions.
Ask a question Ask a questionSearch Answers
Learn more about this topic:
Get access to this video and our entire Q&A library
Washer Method in Calculus | Formula & Equation
from
Chapter 2 / Lesson 16
61K
Define the washer method in calculus. Learn the washer method formula. Discover how to find the volume of a shape using integration and view examples.
Related to this Question
- Find the volume of the solid obtained by rotating the region bounded by y = 1-x^2, y =0 about the x-axis.
- Find the volume of the solid obtained by rotating the region bounded by y = x+2, y = x, x=4 about the x-axis.
- Find the volume of a solid obtained by rotating the region bounded by y= x^2 and x=y^2 about the x-axis.
- Find the volume of the solid obtained by rotating the region bounded by y = x^2 and y = 2x, about the x-axis.
- Find the volume of the solid obtained by rotating the region bounded by y = 1 / x^6, y = 0, x = 3, and x = 9, about the x-axis.
- Find the volume of the solid obtained by rotating the region bounded by y = 4x - x^2, y = x about the x-axis.
- Find the volume of the solid obtained by rotating the region bounded by y = 2\sqrt[4]x, y =2x about the x-axis.
- Find the volume of the solid obtained by rotating the region bounded by y = \sqrt{2x}, y = \sqrt{2-x} about the x-axis.
- Find the volume of the solid obtained by rotating the region bounded by x = \sqrt y, x=y about the x-axis.
- Find the volume of the solid obtained by rotating the region bounded by y = x*sqrt(x), x = 1, x = 3, y = 0 around the x-axis.
- Find the volume of the solid obtained by rotating the region bounded by y = x^4 and y = 1, with x between 0 and 1, around the x-axis.
- Find the volume of the solid obtained by rotating the region bounded by y = \cos x, y = 0, x = 0 about the x-axis.
- Find the volume of the solid obtained by rotating the region bounded by y = \frac{1}{2}x, y=0, x=0, x=1 about the x-axis.
- 4. Find the volume of the solid obtained by rotating the region bounded by y = x^2, y = 0, and x = 3 about the x-axis.
- Find the volume of the solid obtained by rotating the region bounded by y = 10x and y = 2x^2 around the x-axis.
- Find the volume of the solid obtained by rotating the region bounded by y = 5x^2 , x = 1 , and y = 0 about the x-axis.
- Find the volume of the solid obtained by rotating the region bounded by y = 5x^4, y = 5x ; \quad x \geq 0 about the x-axis.
- Find the volume of the solid obtained by rotating the region bounded by y =\frac{x^2}{4} and y = 5-x^2 about the x-axis.
- Find the volume of the solid obtained by rotating the region bounded by f(x) = 2x^2, y = 2, x= 2 about the x-axis.
- Find the volume of the solid obtained by rotating the region bounded by y^2=8x, x=2, x=0, y=0 about the x-axis.
- Find the volume of the solid obtained by rotating the region bounded by y = sin^{-1}x, x = 1 and y = 0 about the x-axis.
- Find the volume of the solid obtained by rotating the region bounded by y = 3-x^2, x=1, x=0 about a. the x-axis. b. the y-axis.
- Find the volume of the solid obtained by rotating the region bounded by y = sqrt{x}, x = 4, y = 2, and the y-axis, about the x-axis.
- Find the volume of the solid obtained by rotating the region bounded by y = x^3, y = 1, and the y-axis around the x-axis.
- Find the volume of the solid obtained by rotating the region above the x-axis bounded by y = x and y = x^7 about the y-axis.
- Find the volume of the solid obtained by rotating the region bounded by y =x \enspace and \enspace y = x^2 about the x-axis.
- Find the volume of the solid obtained by rotating the region bounded by the curves y = 9x^3, y = 9x, x \gt 0 about the x-axis
- Find the volume of the solid obtained by rotating the region bounded by the following curves about the x-axis. y = 16 - x, y = 13x + 12, x = -1.
- Find the volume of the solid obtained by rotating the region bounded by the curves y = e^x, y = 0, x = -1, and x = 1 about the x-axis.
- Find the volume of the solid obtained by rotating the region bounded by the curves y = x^2 and y = x about the x-axis.
- Find the volume of the solid obtained by rotating the region bounded by the curves x = -y^2 + 4y and y = x about the x-axis.
- Find the volume of the solid obtained by rotating the region bounded by the curves x+y=1, x = 2-(y-1)^2 about the x-axis.
- Find the volume of the solid obtained by rotating the region bounded by the curves y = sqrt{x - 1}, y = 0, and x = 5 about the x-axis.
- Find the volume of the solid obtained by rotating the region bounded by the curves y = x 2 , x = y 2 about the x-axis.
- Find the volume of the solid obtained by rotating the region bounded by the curves y = x^2 , y = 9 , x = 0 about the x-axis.
- Find the volume of the solid obtained by rotating the region bounded by the curves y = x^2 and y = 4x about the x-axis.
- Find the volume of the solid obtained by rotating the region bounded by the curves y = 1/x, y = 0, x = 1, and x = 4 about the x-axis.
- Find the volume of the solid obtained by rotating the region bounded by y= x^2, y=0, and x=3 about the x-axis.
- Find the volume of the solid obtained by rotating the region bounded by y = 4 x^2, x = 1, x = 4 and y = 0, about the x-axis.
- Find the volume of the solid obtained by rotating the region bounded by the curve y = 4x - x^2, x = 0 and x = 4 about the x-axis.
- Find the volume of the solid obtained by rotating the region bound by y = \frac{2}{x}, x=1,x=4, y=0 about the x-axis.
- Find the volume of the solid obtained by rotating the region bound by y = \sqrt{36-x^2}, y=0 about the x-axis.
- Find the volume of the solid obtained by rotating the region bound by y = x^2, x=1, x=4, y =0 about the x-axis.
- Find the volume of the solid obtained by rotating the region bound by y = 3\sqrt{\sin x} , y=3 about the x-axis.
- Find the volume of the solid obtained by rotating the region bound by y = e^{x-1} , y=0,x=1, x=4 about the x-axis.
- Find the volume of the solid obtained by rotating the region bound by y = 3x^2, x=1, y=0 about the x-axis.
- Find the volume of the solid obtained by rotating the region bounded by y = x and y = x 2 (a) about the x-axis, and (b) about the line y = 2 .
- Find the volume of the solid obtained by rotating about the x-axis the region bounded the curve y = 1 / x and the x-axis between x = 7 and x = 9.
- Find the volume of the solid obtained by rotating about the x-axis the region bounded by the curve y = 1/x and the x-axis between x = 7 and x = 9.
- Find the volume of the solid obtained by rotating the region bounded by the graphs of: a. y = x^2 - 9, y = 0 about the x-axis. b. y = 16 - x, y = 3x + 12, x = -1 about the x-axis. c. y = x^2
- Find the volume of the solid obtained by rotating the region enclosed by y = e^(5x)+1, y = 0, x = 0, x = 0.3 about the x-axis.
- Find the volume of the solid obtained by rotating the region enclosed by x = 0, x = 1, y = 0, y= = 7 + x^6 about the x-axis.
- Find the volume of the solid obtained by rotating the region bounded by y = \frac{8}{\sqrt{8x-x^2, x=1, x=7 about the x-axis.
- Find the volume of the solid obtained by rotating the region bounded by y= x^2, y=0, x=0, and x=3 about the y-axis.
- Find the volume of the solid obtained by rotating about the x-axis the region bounded by the curves y = e^(-2x), y = 1 + x, and x = 1.
- Find the volume of the solid obtained by rotating about the x-axis the region bounded by the curves y = e^t, y = 0, x = 0, and x = 1.
- Find the volume of the solid obtained by rotating about the x-axis the region bounded by the curve y=\frac{1}{x} and the x-axis between x=7 and x=9.
- Find the volume of the solid obtained by rotating the finite region bounded by y = x^4 and y = 8 x about the x-axis.
- Find the volume of the solid obtained by rotating the finite region bounded by y = x^4 and y = 8x about the x-axis.
- Find the volume of the solid obtained by rotating the region bounded by y = 7x - 10, y =10-3x , x =0 about the y-axis.
- Find the volume of the solid obtained by rotating the region bounded by x = y^2 and x = 4 around the y-axis.
- Find the volume of the solid obtained by rotating the region bounded by y = cos (2x) - x, y = x^2, x = 0, and x = -pi/4 about the x-axis.
- Find the volume of the solid obtained by rotating the region bounded by y = \ln 3x, y=2, y =5, x=0 about the y-axis.
- Find the volume of the solid obtained by rotating the region bounded by y = x^2, x = 0, y =4 about the y-axis.
- Find the volume of the solid obtained by rotating the region bounded by x = 1, x = 2, y = 0, and y = {4 x^2 - 1} / {4 x^4 + x^2} about the y - axis.
- Find the volume of the solid obtained by rotating the region bounded by y = \frac{3}{10}x^2, y = \frac{13}{10} - x^2 about the x-axis.
- Find the volume of the solid obtained by rotating the region bounded by \hspace{10mm} y = \frac{1}{x^4}, \; y= 0, \; zx= 1, \; x = 6 about the y -axis.
- Find the volume of the solid obtained by rotating the region bounded by y = 6x + 54, y = 0 about the y-axis.
- Find the volume of the solid obtained by rotating the region bounded by y = e^{3x} +1, y=0,x=0, x=0.9 about the y -axis.
- Find the volume of the solid obtained by rotating the region bounded by y = 2x(x-2)^2, x=0, y=0 about the y-axis.
- Find the volume of the solid obtained by rotating the region bounded by y = 2x + 1, x = 0, x = 2, and y = 0 around the x-axis.
- Find the volume of the solid obtained by rotating the region bounded by y = 5x(x-2)^2 about the y-axis.
- Find the volume of the solid obtained by rotating the region bounded by y = \displaystyle \frac{1}{x^4} , \ y =0, \ x = 1 , \ x = 6 about the y-axis.
- Find the volume of the solid obtained by rotating the region bounded by y = 5\sqrt x and y = 8x^5 about the y-axis.
- Find the volume of the solid obtained by rotating the region bounded by y = 4 + 3x - x^2, x+y=4 about the y-axis.
- Find the volume of the solid obtained by rotating the region bounded by y = x^2 + 18, y=x^2, x=0 about the y-axis.
- Find the volume of the solid obtained by rotating the region bounded by y=x^2 , y=0, and x=1 , about the y-axis.
- Find the volume of a solid obtained by rotating the region bounded by y = x^2, x = 2y, about the y-axis.
- Find the volume of the solid obtained by rotating the region bounded by y = 6x^2, x =1, y = 0 about the x-axis.
- Find the volume of the solid obtained by rotating the region bounded by y = x - x^2 and y = 0 about the y-axis.
- Find the volume of the solid obtained by rotating the region bounded by y = x^3, y = 8 and x = 0 about the y-axis.
- Find the volume of the solid obtained by rotating the region bounded by y = \frac{x^2}{36}, x=5, y =0 about the y-axis.
- Find the volume of the solid obtained by rotating the region bounded by x=\frac{5}{6}y, \ y=6, \text{ and } x=0 about the y axis.
- Find the volume of the solid obtained by rotating the region bounded by y = x6 and y = x2 about the y-axis.
- Find the volume of the solid obtained by rotating the region bounded by x = 3 and y = 3 - (x^2)/9 about the y-axis.
- Find the volume of the solid obtained by rotating the region bounded by x=7y^2, \ y=1, \ x=0, about the y-axis.
- Find the volume of the solid obtained by rotating the region bounded by y = \sqrt{x-2} , y=0,x=6 about the y-axis.
- Find the volume of the solid obtained by rotating the region bounded by y = 9-x^2, x=0, y=0 about the x-axis.
- Find the volume of the solid obtained by rotating the region bounded by y = 2+ \ln x, y =1, y =2, x=0 about the y-axis.
- Find the volume of the solid obtained by rotating the region bounded by y = 1 + \frac{1}{x^4}, y=1, x=1, x=7 about the x-axis.
- Find the volume of the solid obtained by rotating the region bounded by y=(1)/(x^5), y=0, x=3, and x=5, about the y-axis.
- Find the volume of the solid obtained by rotating the region bounded by y = 2\sqrt{\sin x}, y=0, x=1 about the x-axis.
- Find the volume of the solid obtained by rotating the region bounded by x=y^2 and x=2y about the y-axis.
- Find the volume of the solid obtained by rotating the region bounded by x=y-y^2 and x=0, about the y-axis.
- Find the volume of the solid obtained by rotating the region bounded by x = 2y, y = 2, y = 3, x = 0 about the x-axis.
- Find the volume of the solid obtained by rotating the region bounded by y = 1/(x^2 + 1), x = 0, x = 3, y = 0, about the y-axis.
- Find the volume V of the solid obtained by rotating the region bounded by y^2 = 2x \text{ and } x=2y about the y axis.
- Find the volume of the solid obtained by rotating the region bounded by y = 8x + 56, y = 0, about the y-axis.
- Find the volume of the solid obtained by rotating the region bounded by y^2 = 3x, x = 3y about the y-axis.
- Find the volume of the solid obtained by rotating the region bounded by y=3sin(7x^2), y=0, 0 leq x leq sqrtpi 7 , about the y-axis.
Explore our homework questions and answers library
Browse
by subject