![]() ![]() This entry was posted in Introductory Problems, Volumes by cross-section on Jby mh225. The cross-sections are circles of radius x 2, so the cross-sectional area is A(x) π⋅(x 2) 2π⋅x 4 The volume is V = ∫ -1 1A(x) dx = ∫ -1 1 π⋅x 4 dx = π⋅(x 5/5)| -1 1 = 2π/5 Find the volume of the solid obtained by rotating the curve y = x 2, -1 ≤ x ≤ 1, about the x-axis. where x⋅ex 2 was integrated using the substitution u = x 2, so du = 2xdx.ĥ. The area is A(x) = base ⋅ height = x⋅ex 2. Cross sections that are perpendicular to the x-axis are equilateral triangles. Find the volume of the solid with cross-section a rectangle of base x and height e x 2 Answerġ. isosceles right triangles with the legs perpendicular to the x-axis. where cos(x)sin 2(x) is integrated using the substitution u = sin(x), so du = cos(x) dx.Ĥ. ![]() Cross Section: Semicircles, diameter in the xy-plane. Find the volume of the solid with circular cross-section of radius cos 3/2(x), for 0 ≤ x ≤ π/2. Cross Section: Isosceles Right Triangles, one of the legs in the xy-plane. Recall an ellipse with semi-major axis a and semi-minor axis b has area πab, so this ellipse with semi-major axis x 2 and semi-minor axis x 3 has the area: A(x) = π⋅x 2⋅x 3 = π⋅x 5. Find the volume if the solid with elliptical cross-section perpendicular to the x-axis, with semi-major axis x 2 and semi-minor axis x 3, for 0 ≤ x ≤ 1 Answerġ. Find the volume of the solid with right isosceles triangular cross-section perpendicular to the x-axis, with base x 2, for 0 ≤ x ≤ 1 Answerġ. ![]() It worked great! Now each cross-section stands nicely spaced and vertical.1. Then I carefully slid each cross-section into its appropriate slit. Update: I used spray adhesive to glue the base area to some foamboard and cut slits in it with an Xacto knife. You can download the templates provided by Nina Otterson here. That way, they will stand up straighter and stay evenly spaced. When I do this activity next year, I think I’ll glue the base area to foamboard, and have students insert the cross-sections into slits cut into the foamboard. Find the volume if the solid with elliptical cross-section perpendicular to the x-axis, with semi-major axis x 2 and semi-minor axis x 3, for 0 x 1 Answer Solution 3. Building a model using actual cross-sections made all the difference! Find the volume of the solid with right isosceles triangular cross-section perpendicular to the x-axis, with base x 2, for 0 x 1 Answer Solution 2. I’ve never had students grasp the idea behind this type of volume as quickly and as easily as this group did. Once they understood that the thickness of the paper was dx, it was very easy to set up the integrals to calculate the volumes of their models. For this solid, each cross section perpendicular to the x-axis is an isosceles right triangle with a leg in R. Here they are, taping the cross-sections onto the base area: Here are my students in action, cutting out the cross-sections: Students use the templates to cut out a cross-section that fits down the middle of the base area, and six others on each side. I had my students cut the square templates diagonally for isosceles right triangles, and horizontally for rectangles. In her session, Nina Otterson provided templates that fit the given base area for different shapes: semicircles, squares, and equilateral triangles. Here’s what the base area looks like, courtesy of ’s online function grapher: She has her students cut cross-sections of different shapes and apply them to a base area enclosed by two parabolas, y = x^2 – 3 and y = 3 – x^2. Nina Chung Otterson was the presenter, and she teaches at The Hotchkiss School in Connecticut. Last year, I went to the regional NCTM conference here in Nashville, TN, and one of the sessions I attended addressed this exact issue. It’s hard, because they have difficulty visualizing it. One of the hardest type of problem for calculus students to understand is calculating the volume of solids of known cross-sections. ![]()
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