Happy holidays, everyone! Hope you're having a great start to break! Today, I am going to be elaborating more on another important concept in calculus known as the disk method.
**NOTE: the shape extends from 0 to 1 (along x-axis) so the shape ends at 1** In order to find the volume of the shaded region, we have to subtract the volume of the cone shape from the volume of the dome shape. As per the solid of revolution formula, we know that V = π∫(b to a) [R(x)]^2 dx.
π∫(1 to 0)(x^1/2)^2 dx - π∫(1 to 0)(x)^2 dx π∫(1 to 0)x dx - π∫(1 to 0) x^2 dx = π[1/2 x^2] - π[1/3 x^3] = (substitue 1 and 0) 1/2 π - 1/3 π (3-2 π)/6 = 1/6π So the volume of the purple area is 1/6 π!! With this method, you can practically find the volume of any distinct 3 dimensional shape! Very interesting, right? Good luck and happy holidays,
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