Applications of Integration

Question 1

Find the mean of the function f(x)

Answer 1

= 1/(b-a) ∫f(x) dx

Upper bound = b
Lower bound = a

Question 2

Find the root mean square of the function f(x)

Answer 2

= ( 1/(b-a) ∫f^2(x) dx )^1/2

same as mean just f(x) is squared, while everything is rooted
Upper bound = b
Lower bound = a

Question 3

Find the length of the curve

Answer 3

= ∫ ( 1 + (dy/dx)^2 )^1/2
= ∫ of all the gradients

Question 4

Find the length of the curve using polar coordinates

Answer 4

= ∫ ( r^2 + (dr/dθ)^2 )^1/2

Question 5

Calculate the surface area of the function f(x)

Answer 5

= ∫ 2πy ( 1 + (dy/dx)^2 )^1/2

Length = 2πy
Width = (dx^2 + dy^2)^1/2

Question 6

Calculate the volume of the function f(x) as it rotates around the x axis

Answer 6

V = ∫ π y^2 dx