Application of Integrals Flashcards
Explain how to solve for the Volume of solid S
Start by acknowledging the function x² +y² = 16.
Since the Volume is taken using semi-circle cross-section perpendicular to the x axis, we have set this function with respect to y and solve for the two x - intercepts
equation: y = √(16 - x²)
x-intercept = [-4,4] This will be our bounds
now to find the area of the cross sectional cut we need to understand that the radius will be the equation with respect to y. this will make our area:
A = [π/2]r² = [π/2][√(16 - x²)]² = [π/2][16 - x²]
now we can take the integral with respect to x from -4 to 4:
∫ [π/2][16 - x²] dx; from [-4,4]
What is the equation to Volume by Slicing of any function
∫A(x)dx from x = [a,b]
where A(x) is the cross-section of the arbitrary shape used to define the volume.
What is the general Shell Method formula for Solids of Revolution ?
∫ 2π R(x)H(x)dx
Where:
R(x) = distance of object from axis of rotation
H(x) = the general thickness of the object with respect to the functions generating the object
What is the equation to the Disk method and how is it derived ?
∫ πr² d(x,y); where
*r**, radius of the circle created = the function being rotated
d(x,y) = the range which the volume is being taken
This comes from the fact that rotating an object around an object will typically create a spheric cross section having the axis as the normal radius.
Find the volume of the solid generated when the region bounded by y = 4x and
y = 8√(x) is revolved about the x - axis.
Explain steps to solve this volume
1. Start with choosing the type of volume method we will use: since this is revolved around the x axis, and our equations are already with respect to x, we will use Disc Method
2. Identify the equation: ∫ π (R² - r²)dx Where R: is 8√(x) & r = 4x
3. Identify the bounds by setting up 8√(x) = 4x so x = (0,4)
4. Now set up our integral: ∫ π [(8√(x))² - (4x)²]dx from x = (0,4)
Explain the solution to this:
1. Disk Method can be used for this as it is with respect to x and about the x axis.
2. Identify the shape of this graph and equation. Since the 18√(sinx) is at 0 at x = 0, we can assume that it is the lower part of this graph and can be treated as r
∫ π (R² - r²)dx
3. Then the bounds will be set using 18√(sinx) = 18 so x = (0,π/2)
4. Now set up our integral: ∫ π [(18)² - (18√(sinx))²]dx from x = (0,π/2)
How to find the volume of this revolved around the y axis using the shell method ?
Since we are revolving around the y axis using shell method, all our radius measures will be done x as our variable.
1. Start by identifying the distance of our solid from the axis of rotation: x
2. Then define out thickness of our object using the bounds set by the two functions: L = 13x - x²
3. Now load it into our shell function: ∫ 2π⋅x(13x - x²)dx
How do we solve the volume of this using Shell Method ?
1. We identify that this object is restricted by two functions and an axis: y = 33x, and y = 33
2. we identify our length as x since it is just on the x axis
3. we identify our thickness as wrt x as (33 - 33x) since it is our top function minus our bottom.
4. our function will therefore be ∫ 2π⋅x(33 - 33x)dx
How do we find the volume of this solid revolved around the X axis using Shell method ?
1. Since we’re revolving around the x axis we are going to do everything with respect to y
2. we will identify our length as y
3. we will change our function to be with respect to y, x = y², and our thickness will be set to (49 - y²)
4. Set up our function as ∫ 2π⋅y(49 - y²)dy
What is the equation for arch length ?
Arch = ∫ √(1 + ƒ’(x)²) dx
