A bit messy The easy way is to note that r r. We end up integrating r2cos2 x r 2 cos 2 x, and use the double angle formula cos 2 2cos2 1 cos 2 2 cos 2 1. The diameter of the semicircle is determined by a point on the line x + 4 y = 4 and a point on the x‐axis (Figure 2). The conventional way to find r r r2 x2 dx r r r 2 x 2 d x is to make the trig substitution x r sin x r sin. The area ( A) of an arbitrary square cross section is A = s 2, whereĮxample 2: Find the volume of the solid whose base is the region bounded by the lines x + 4 y = 4, x = 0, and y = 0, if the cross sections taken perpendicular to the x‐axis are semicircles.īecause the cross sections are semicircles perpendicular to the x‐axis, the area of each cross section should be expressed as a function of x. The length of the side of the square is determined by two points on the circle x 2 + y 2 = 9 (Figure 1). In this case, the volume ( V) of the solid on isĮxample 1: Find the volume of the solid whose base is the region inside the circle x 2 + y 2 = 9 if cross sections taken perpendicular to the y‐axis are squares.īecause the cross sections are squares perpendicular to the y‐axis, the area of each cross section should be expressed as a function of y. If the cross sections are perpendicular to the y‐axis, then their areas will be functions of y, denoted by A(y). The volume ( V) of the solid on the interval is If the cross sections generated are perpendicular to the x‐axis, then their areas will be functions of x, denoted by A(x). You can use the definite integral to find the volume of a solid with specific cross sections on an interval, provided you know a formula for the region determined by each cross section. Volumes of Solids with Known Cross Sections Volumes of Solids with Known Cross Sections.Second Derivative Test for Local Extrema.First Derivative Test for Local Extrema.Isosceles Right Triangle: The volume of the isosceles. Differentiation of Exponential and Logarithmic Functions Cross-sections perpendicular to the x-axis are isosceles right triangles with hypotenuse in the base. Differentiation of Inverse Trigonometric Functions.Limits Involving Trigonometric Functions.
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