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Let R be the region in the first quadrant bounded by the graph of y = e^x , the…

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Wize University Calculus 1 Textbook > Applications of Integration

Volumes of Revolution by Cross Sections (Washers)

5 Activities

Let RR be the region in the first quadrant bounded by the graph of y=exy = e^x , the line y=2y = 2 and the yy-axis; let SS be the solid of revolution obtained by revolving RR about the xx-axis. Compute the volume of SS with the disc/washer method.

More Volumes of Revolution by Cross Sections (Washers) Questions:
Volumes of Revolution by Cross Sections
Find the volume of the solid obtained by rotating the region bounded by the curves y=5−x2y=5-x^2 and y=4y=4 about the line 𝑦 = 2.
Volumes of Revolution by Cross Sections
Find the integrals that represents the volume of the solid obtained by rotating the region bounded by y=2xy=\frac{2}{\sqrt{x}}, y=1y=1, and x=1x=1
a.) around the xx-axis
b.) around the yy-axis
Write the integrals representing the volume of the solid that is produced by revolving the region bounded between y=tan⁡3xy=\tan^3x, y=1y=1, and x=0x=0 about the line y=2y=2 using both the washer method and the method of cylindrical shells. Do not compute the integrals.
Find the volume of the solid that is produced by revolving the region bounded by the x-axis and y=xsin⁡xcos⁡xy=\sqrt{x\sin x\cos x} about the x-axis, between 0 and π/2\pi/2.
Write an integral representing the volume of the solid that is produced by revolving the region bounded between y=xy=x and y=3x−x2y=3x-x^2 about the line y=−1y=-1. Do not compute the integral.
Find the volume of the solid that is produced by revolving y=xsin⁡xcos⁡xy=\sqrt{x\sin x\cos x} about the xx-axis, between 0 and π/2\pi/2.
Find the volume of the solid that is produced by revolving y=xsin⁡xcos⁡xy=\sqrt{x\sin x\cos x} about the xx-axis, between 0 and π/2\pi/2.
Find the volume of the solid that is produced by revolving y=xsin⁡xcos⁡xy=\sqrt{x\sin x\cos x} about the xx-axis, between 0 and π/2\pi/2.
Let RR be the region in the first quadrant bounded by the graph of y=exy = e^x , the line y=2y = 2 and the yy-axis; let SS be the solid of revolution obtained by revolving RR about the xx-axis. Compute the volume of SS with the disc/washer method.
Volumes of Revolution by Cross Sections
Find the volume of the solid obtained by rotating the region bounded by the curves y=5−x2y=5-x^2 and y=4y=4 about the line 𝑦 = 2.
Volumes of Revolution by Cross Sections
Find the integrals that represents the volume of the solid obtained by rotating the region bounded by y=2xy=\frac{2}{\sqrt{x}}, y=1y=1, and x=1x=1
a.) around the xx-axis
b.) around the yy-axis
Q:\bf{Q:} Find the volume of the solid obtained by rotating the region bounded by the curves y=5−x2y=5-x^2 and y=4y=4 about the line y=2y=2.
Find the volume of the solid obtained by rotating the region bounded by the curves y=exy=e^x , y=1y=1 and x=4x=4, about the xx-axis and then about the yy-axis. (For the first part, enter the answer. For the second part enter the integral only, do not solve again. Factor out the pi.)
Write the integral that represents the volume of the solid generated by revolving the region bounded by y=x3y=\sqrt[3]{x} and y=x4y=\dfrac{x}{4} that lies in the first quadrant, about the yy-axis. Use the method of disks/washers. Do not solve.
Volumes of Revolution by Cross Sections
Find the volume of the solid obtained by rotating the region bounded by the curves y=5−x2y=5-x^2 and y=4y=4 about the line 𝑦 = 2.
Volumes of Revolution by Cross Sections
Find the integrals that represents the volume of the solid obtained by rotating the region bounded by y=2xy=\frac{2}{\sqrt{x}}, y=1y=1, and x=1x=1
a.) around the xx-axis
b.) around the yy-axis
Write an integral representing the volume of the solid that is produced by revolving the region bounded between y=xy=x and y=3x−x2y=3x-x^2 about the line y=−1y=-1. Do not compute the integral.
Practice: Volume of Cone
Q:\textbf{Q:} Derive the formula for the volume of a right circular cone whose altitude is hh and whose radius of the base is rr.
Practice: Rotate about Horizontal Line
Q:\textbf{Q:} Find the volume of the solid generated when the region enclosed by the curves y=x2y = x^2 and y=x+2y = x + 2 is revolved about the line y=−1. y = −1.
Practice: Rotate about y-axis
Q:\textbf{Q:} Consider the region RR of the first quadrant bounded by the curves y2=xy^2 = x and y=x4y = x^4. Find the volume of the region created by revolving R R around the y-axis. Use the method of cross-sections.
Practice: Rotate about x-axis
Q:\textbf{Q:} Find the volume of the solid obtained by revolving the region enclosed by y=4−5x22y=4-\dfrac{5x^2}{2} and y=x22+1y=\dfrac{x^2}{2}+1 around the xx-axis.
Practice: Rotate about x-axis
Q:\textbf{Q:} Find the volume of the solid obtained by revolving the region between y=x2+1y=x^2+1 and y=3−xy=3-x around the xx-axis.
Practice: Volumes of Revolution (~F2018 Final Q43)
Practice: Volumes of Revolution
Find the integrals that represents the volume of the solid obtained by rotating the region bounded by y=2xy=\frac{2}{\sqrt{x}}, y=1y=1, and x=1x=1
a.) around the xx-axis
volume around y=2
Find the volume of the solid obtained by rotating the region bounded by the curves y=5−x2y=5-x^2 and y=4y=4 about the line 𝑦 = 2.
volume around y=2
Find the volume of the solid obtained by rotating the region bounded by the curves y=5−x2y=5-x^2 and y=4y=4 about the line 𝑦 = 2.
volume around y=2
Find the volume of the solid obtained by rotating the region bounded by the curves y=5−x2y=5-x^2 and y=4y=4 about the line 𝑦 = 2.
volume around y=2
Find the volume of the solid obtained by rotating the region bounded by the curves y=5−x2y=5-x^2 and y=4y=4 about the line 𝑦 = 2.
volume around y=2
Find the volume of the solid obtained by rotating the region bounded by the curves y=5−x2y=5-x^2 and y=4y=4 about the line 𝑦 = 2.
volume around y=2
Find the volume of the solid obtained by rotating the region bounded by the curves y=5−x2y=5-x^2 and y=4y=4 about the line 𝑦 = 2.
volume around y=2
Find the volume of the solid obtained by rotating the region bounded by the curves y=5−x2y=5-x^2 and y=4y=4 about the line 𝑦 = 2.
volume around y=2
Find the volume of the solid obtained by rotating the region bounded by the curves y=5−x2y=5-x^2 and y=4y=4 about the line 𝑦 = 2.
volume around y=2
Find the volume of the solid obtained by rotating the region bounded by the curves y=5−x2y=5-x^2 and y=4y=4 about the line 𝑦 = 2.
volume around y=2
Find the volume of the solid obtained by rotating the region bounded by the curves y=5−x2y=5-x^2 and y=4y=4 about the line 𝑦 = 2.
volume around y=2
Find the volume of the solid obtained by rotating the region bounded by the curves y=5−x2y=5-x^2 and y=4y=4 about the line 𝑦 = 2.
volume around y
Find the volume of the solid obtained by rotating the region bounded by the curves y=xy=\sqrt{x} and x=3yx=3y around the 𝑦-axis.
volume around y
Find the volume of the solid generated by rotating the region bounded by y=x2y=x^2 and y=xy=x about the yy-axis.
(a) π3\frac{\pi}{3} (b) π12\frac{\pi}{12} (c) π2\frac{\pi}{2} (d) 3π4\frac{3\pi}{4} (e) π6\frac{\pi}{6}
Write the integrals representing the volume of the solid that is produced by revolving the region bounded between y=tan⁡3xy=\tan^3x, y=1y=1, and x=0x=0 about the line y=2y=2 using both the washer method and the method of cylindrical shells. Do not compute the integrals.
Find the volume of the solid that is produced by revolving the region bounded by the x-axis and y=xsin⁡xcos⁡xy=\sqrt{x\sin x\cos x} about the x-axis, between 0 and π/2\pi/2.
Write an integral representing the volume of the solid that is produced by revolving the region bounded between y=xy=x and y=3x−x2y=3x-x^2 about the line y=−1y=-1. Do not compute the integral.
Volumes of Revolution by Cross Sections
Evaluate the volume obtained by rotating the region enclosed by the curves y=ex,  y=1,  x=ey=e^x,\ \ y=1,\ \ x=e around the x-axis
Write the integral that represents the volume of the solid generated by revolving the region bounded by y=x3y=\sqrt[3]{x} and y=x4y=\dfrac{x}{4} that lies in the first quadrant, about the yy-axis. Use the method of disks/washers. Do not solve.
Write the integrals representing the volume of the solid that is produced by revolving the region bounded between y=tan⁡3xy=\tan^3x, y=1y=1, and x=0x=0 about the line y=2y=2 using both the washer method and the method of cylindrical shells. Do not compute the integrals.
Find the volume of the solid that is produced by revolving y=xsin⁡xcos⁡xy=\sqrt{x\sin x\cos x} about the xx-axis, between 0 and π/2\pi/2.
Write an integral representing the volume of the solid that is produced by revolving the region bounded between y=xy=x and y=3x−x2y=3x-x^2 about the line y=−1y=-1. Do not compute the integral.
Write an integral representing the volume of the solid that is produced by revolving the region bounded between y=xy=x and y=3x−x2y=3x-x^2 about the line y=−1y=-1. Do not compute the integral.
Find the volume of the solid obtained by rotating the region bounded by x=5/yx=5/y , y=1y=1 and x=2x=2 about the y-axis.
volumeint
Write the integral that represents the volume of the solid obtained by rotating the region bounded by y=5xy=\frac{5}{x} , y=1y=1 and x=2x=2 about the x-axis.
Write the integral that represents the volume of the solid generated by revolving the region bounded by y=x3y=\sqrt[3]{x} and y=x4y=\dfrac{x}{4} that lies in the first quadrant, about the yy-axis. Use the method of disks/washers. Do not solve.
Q:\bf{Q:} Find the volume of the solid obtained by rotating the region bounded by the curves y=5−x2y=5-x^2 and y=4y=4 about the line y=2y=2.
Find the volume of the solid obtained by rotating the region bounded by the curves y=exy=e^x , y=1y=1 and x=4x=4, about the xx-axis and then about the yy-axis. (For the first part, enter the answer. For the second part enter the integral only, do not solve again. Factor out the pi.)
Practice: Rotate about x-axis
Q:\textbf{Q:} Find the volume of the solid obtained by revolving the region between y=x2+1y=x^2+1 and y=3−xy=3-x around the xx-axis.
Practice: Rotate about Horizontal Line
Q:\textbf{Q:} Find the volume of the solid generated when the region enclosed by the curves y=x2y = x^2 and y=x+2y = x + 2 is revolved about the line y=−1. y = −1.
Practice: Rotate about y-axis
Q:\textbf{Q:} Consider the region RR of the first quadrant bounded by the curves y2=xy^2 = x and y=x4y = x^4. Find the volume of the region created by revolving R R around the y-axis. Use the method of cross-sections.
Practice: Volume of Cone
Q:\textbf{Q:} Derive the formula for the volume of a right circular cone whose altitude is hh and whose radius of the base is rr.
Practice: Rotate about x-axis
Q:\textbf{Q:} Find the volume of the solid obtained by revolving the region enclosed by y=4−5x22y=4-\dfrac{5x^2}{2} and y=x22+1y=\dfrac{x^2}{2}+1 around the xx-axis.
Write the integrals representing the volume of the solid that is produced by revolving the region bounded between y=exy=e^x and y=(e−1)x+1y=(e-1)x+1 about the line y=−1y=-1 using both the washer method and the method of cylindrical shells. Do not compute the integrals.
Write an integral representing the volume of the solid that is produced by revolving the region bounded between y=xy=x and y=3x−x2y=3x-x^2 about the line y=−1y=-1. Do not compute the integral.
Write the integrals representing the volume of the solid that is produced by revolving the region bounded between y=tan⁡3xy=\tan^3x, y=1y=1, and x=0x=0 about the line y=2y=2 using both the washer method and the method of cylindrical shells. Do not compute the integrals.
Compute the volume of the solid formed when the area between f(x)=x2+1f\left(x\right)=x^2+1 , x=0x=0, x=2x=2 , and y=0y=0 is rotated around the y-axis.
Write the integral that represents the volume of the solid obtained by rotating the region bounded by 𝑦 = ln 𝑥 , 𝑦 = 2 and 𝑥 = 10 around the 𝑥-axis. Do not solve.
Write the integral that represents the volume of the solid generated by revolving the region bounded by 𝑦 =3x3\sqrt{x} and 𝑦 = x4\frac{x}{4} that lies in the first quadrant (top-right corner) about the 𝑦-axis. Do not solve.
Find the volume of the solid that is produced by revolving y=xsin⁡xcos⁡xy=\sqrt{x\sin x\cos x} about the xx-axis, between 0 and π/2\pi/2.
Volumes of Revolution by Cross Sections
Find the volume of the solid obtained by rotating the region bounded by the curves y=5−x2y=5-x^2 and y=4y=4 about the line 𝑦 = 2.
Volumes of Revolution by Cross Sections
Find the integrals that represents the volume of the solid obtained by rotating the region bounded by y=2xy=\frac{2}{\sqrt{x}}, y=1y=1, and x=1x=1
a.) around the xx-axis
b.) around the yy-axis
volume around y
Find the volume of the solid obtained by rotating the region bounded by the curves y=xy=\sqrt{x} and x=3yx=3y around the 𝑦-axis.
Volumes of Revolution by Cross Sections
Find the volume of the solid generated by rotating the region bounded by y=x2y=x^2 and y=xy=x about the yy-axis.
Volumes of Revolution by Cross Sections
Which of the following integrals represent the volume of the solid generated by revolving the area bounded by y=x+2y=x+2 , y=x2y=x^2 andx=0x=0 around the y-axis.