Flashcards
Volumes of cylinders and cones
Flashcards do conjunto (14)
- 1How do we find the volume of a cylinder?π x r^2 x h
- 2What is the formula involving the summation for volume of a cone?π x lim(δ->0) x (b)Σ(a) x y^2 x δx
- 3What do a and b represent in this formula?The limits of the x-value
- 4What does δx represent in this formula?The thickness of the rectangles we are using to estimate the volume of the cone
- 5Why should δx be as low as possible?It will give a more accurate volume
- 6Ways to improve our volume estimation are:Increasing number of rectangles and increasing number of cylinders
- 7Image to visualize it:
- 8This imagine represents:Rectangles along x-axis rotation by 360 degrees (revolution)
- 9As δx approaches 0, the sum becomes an....Integral
- 10Another way to find the volume of a cone rotated by 360 degrees in the x-axis is by using the formula.....π x b∫a (y^2) dx
- 11We can also find the volume of a cone rotated 360 degrees in the y-axis by....π x b∫a (x^2) dx
- 121. Region bounded between x^2 +5 between x=0 and x=4 and the x-axis is rotated 360 degrees about x-axis. Find the volume to 3sfπ x 4∫0 (x^2+5)^2 dx
4∫0 (x^2+5)^2 dx = 518.133
518.133 x π = 1627.76
3sf = 1630 - 132. Region bounded between x^2 +5 between y=6 and y=10 and the y-axis is rotated 360 degrees about the y-axis. Find the volume to 3sfRearrange y=x^2 +5 for x^2=y-5 so we can use formula π x b∫a (x^2) dx
So 10∫6 (y-5)^2 dx = 12
π x 12 = 37.699
3sf = 37.7 - 143. Region bounded by y=3x^2 +1 between x=0 and x=2 and the y-axis is rotated by 360 degrees about the y-axis. Find the volume to 3sfFind y boundaries: 3(0)^2 + 1 = 1
3(2)^2+1 = 13
y=3x^2 +1 rearranged = x^2 = (y-1)/3
Now: π x 13∫1 (y-1)/3 dx = 75.4