![]() ![]() It is a special isosceles triangle with one angle being a right angle and the other two angles are congruent as the angles are opposite to the equal sides. This entry was posted in Introductory Problems, Volumes by cross-section on Jby mh225. An isosceles right triangle is defined as a right-angled triangle with an equal base and height which are also known as the legs of the triangle. 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. Find the volume of the solid with cross-section a rectangle of base x and height e x 2 Answerġ. where cos(x)sin 2(x) is integrated using the substitution u = sin(x), so du = cos(x) dx.Ĥ. Find the volume of the solid with circular cross-section of radius cos 3/2(x), for 0 ≤ x ≤ π/2. 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. A right isosceles triangle with base x 2 has altitude x 2 and so area A(x) = (1/2)⋅base⋅altitude (1/2)⋅x 2⋅x 2 = (1/2)⋅x 4 Then the volume is: V = ∫ 0 1A(x) dx = ∫01(1/2)⋅x 4dx = x 5/10| 0 1= 1/10.Ģ. Isosceles Scalene Acute Obtuse Right Triangle Practice Problems Triangles can be classified by various properties relating to their angles and sides. ![]() 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. In geometry, all angles measurements are positive, at least in the geometry we are studying on this site. ![]()
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