MATH1051 Lecture 9: Lecture 09
6.2 – Volumes and Average Value
What is the area under the curve again?
●It’s the sum of the area of infinite rectangles under the function
●What is the area of each rectangle?
○(x)Δxf
●Therefore
○rea (x)dxA = ∫
b
a
f
●What then, is the volume of the area under the curve rotated around a line?
● π (R(x) (x)) dxV = ∫
b
a
2−r2
○Because the area of a circle is , we can compile an integral from this by usingrπ2
(x) r(x)R2− 2
Example #1
●Find the volume of a solid with a base of region R between , which and y xy = √x=
has cross-sections perpendicular to the x-axis in the shape of squares
Document Summary
It"s the sum of the area of infinite rectangles under the function. What is the area of each rectangle? f (x) x. What then, is the volume of the area under the curve rotated around a line? b a f (x)dx r 2. , we can compile an integral from this by using. Because the area of a circle is. Find the volume of a solid with a base of region r between y = x and y x. , which has cross-sections perpendicular to the x-axis in the shape of squares. First, find the intersections between the two functions. X = x x = 2 x x2 x = Then think about the area of a square. X is the top function while x is the bottom. Find the volume of a solid with base r of perpendicular to y-axis that are squares y = x and y x with cross-sections.
