Download Math 104 – Calculus 6.1 Volume by Cross-‐sec ons

Math 104 – Calculus 6.1 Volume by Cross-­‐sec:ons Math 104 -­‐ Yu Volume by cross-­‐sec:ons •  Goal: Find the volume of a solid. •  Method: slice the solid into pieces and sum them up. Math 104 -­‐ Yu Volume by cross-­‐sec:ons •  The
Riemann sum
lim
n
X
A(xk ) xk converges to an integral.
k=1
n
X
xk !0
k=1
A(xk ) xk =
Z
b
A(x) dx
a
Math 104 -­‐ Yu Volume by cross-­‐sec:ons Math 104 -­‐ Yu Example Find the volume of the given pyramid, which has a square base of side-­‐
length 3m and height 5m. (Anima:on) Math 104 -­‐ Yu Example The base of the solid is a quarter of a disk of radius 1. The cross-­‐sec:ons by planes perpendicular to the x-­‐axis are squares with one side on the disk. 1)  Find the volume. (Anima:on) 2)  What if the cross-­‐sec:ons are isosceles right triangles with one leg on the quarter disk? 3)  What if the cross-­‐sec:ons are isosceles right triangles with one leg on the quarter disk? 4)  What if the cross-­‐sec:ons are equilateral triangles? Math 104 -­‐ Yu Solids of Revolu:on •  A solid of revolu,on is obtained by rota:ng a plane region about an axis. Math 104 -­‐ Yu Solids of Revolu:on •  One way to calculate its volume is by using cross-­‐sec:ons perpendicular to the rota:on axis. •  First case: no gap between the region and the axis, “Disk Method” Math 104 -­‐ Yu Solid of Revolu:on •  Second case: there is gap between the region and the axis, “Washer Method” Math 104 -­‐ Yu Disk Method •  Disk Method with horizontal axis of rota:on (not necessarily the x-­‐
axis) R(x) = radius as function in x
Area of cross-sections: A(x) = ⇡[R(x)]2
Volume =
Z
b
A(x) dx =
a
Z
b
⇡[R(x)]2 dx
a
Math 104 -­‐ Yu Example Calculate the
p volume of the solid generated by rotating the region between
the curves
y = x and y = 0 about the x-axis.
Math 104 -­‐ Yu Disk Method •  Disk Method with ver:cal axis of rota:on (not necessarily the y-­‐
axis) R(y) = radius as function in y
Area of cross-sections: A(y) = ⇡[R(y)]2
Volume =
Z
b
A(y) dy =
a
Z
b
⇡[R(y)]2 dy
a
Math 104 -­‐ Yu Example Calculate the volume of the solid generated by rotating the region between
the curves
y = x3 , y = 8, and x = 0 about the y-axis
Math 104 -­‐ Yu Example The semi-­‐circle of radius a is revolved around the x-­‐axis to give a sphere. Find its volume. Math 104 -­‐ Yu Washer Method •  Washer Method with horizontal axis of rota:on (not necessarily the x-­‐axis) R(x) = Outer Radius
r(x) = Inner Radius
Z b
Volume =
⇡[R(x)2 r(x)2 ] dx
a
Math 104 -­‐ Yu Washer Method •  Washer Method with ver:cal axis of rota:on (not necessarily the y-­‐
axis) R(y) = Outer Radius
r(y)Z= Inner Radius
b
⇡[R(y)2
Volume =
r(y)2 ] dy
a
Rotate about the y-­‐axis Math 104 -­‐ Yu Example Calculate the
volume of the solid generated by rotating the region between the
p
y =
x and y = x2 around
the y-axis
curves
Math 104 -­‐ Yu Example Calculate the volume of the solid generated by rotating the region between the
curves y = 4 x2 and y = 0 about the line y = 2
Math 104 -­‐ Yu