Help u-substitution, new boundaries question

[arl146]

what do i do if when i am changing the limits/boundaries on my integral because i did a u substitution and the new limits end up being the same number?

EXAMPLE:

integral is of:

4*(sin(theta)^4)*cos(theta) dtheta

the limits are pi/6 to 5*pi/6
i had u= sin(theta) ... both the u lower and upper limits equal 1/2...

the original integral was a double integral in polar coordinates... limits of r was o-2sin(theta) and i already stated the theta's limits. the function is (r^3)*cos(theta)

it doesn't make sense to me to let u=cos(theta) in this case of
"integral is of:
4*(sin(theta)^4)*cos(theta) dtheta"

 

[Mark44]

what do i do if when i am changing the limits/boundaries on my integral because i did a u substitution and the new limits end up being the same number?

EXAMPLE:

integral is of:

4*(sin(theta)^4)*cos(theta) dtheta

the limits are pi/6 to 5*pi/6
i had u= sin(theta) ... both the u lower and upper limits equal 1/2...

the original integral was a double integral in polar coordinates... limits of r was o-2sin(theta) and i already stated the theta's limits. the function is (r^3)*cos(theta)

it doesn't make sense to me to let u=cos(theta) in this case of
"integral is of:
4*(sin(theta)^4)*cos(theta) dtheta"

If you end up with an integral whose limits of integration are the same, the value of your integral is zero.

An alternate approach involves not changing the limits of integration. First, evaluate the indefinite integral to get an antiderivative of 4sin⁴(t)cos(t). Then evaluate your antiderivative at the upper and lower limits of integration, and subtract.

 

[arl146]

i realized that after i played around with it some more. thanks!