Question

A curved edge is cut from a culinderi of radius 3 cm by teo planes. one plane is perpendicular to the axis of the bcylnder. thr secomd plane crrosses the first plane at a 45° angle ath the center of the cylinder. find the volume of wedge​

Answer 1

Answer:

volume of wedeg is 18 sq unit .

see figure, here we have drawn a typical cross sectional area which is perpendicular to x-axis.

from figure it is clear that cross sectional area at x-axis is rectangle.

let height of rectangle is x

then, width of it will be 2√(3² - x²) [ from Pythagoras theorem ]

= 2√(9 - x²)

so, cross sectional area , dA = x(2x√(9 - x²)

as volume is given as V=\int\limits^a_b{A(x)}\,dxV=b∫aA(x)dx

here, A(x) = 2x√(9 - x²) , a = 3 and b = 0

so, V = \int\limits^3_0{(2x\sqrt{9-x^2})}\,dx0∫3(2x9−x2)dx

[using application, \int{x\sqrt{a^2-x^2}}\,dx=-\frac{1}{3}(a^2-x^2)^{3/2}∫xa2−x2dx=−31(a2−x2)3/2 ]

= -\frac{2}{3}\left[(9-x^2)^{3/2}\right]^3_0−32[(9−x2)3/2]03

= -2/3 × [0 - (9)^(3/2)]

= -2/3 × -27

= 18