I understand that while changing an integral from cartesian to cylindrical or polar or spherical coordinates requires multiplication of a Jacobian to integration variables but what when the region is itself expressed in cylindrical system for example
consider cone $z=3r$ , rose petal r=$sin4(\theta)$ and $\theta$ (0,$\pi$/4)
now to integrate this do I write
$$\int_{0}^{\pi/4} \int_{0}^{sin4\theta}\int_0^{3r} dz\ dr\ d\theta$$
or
$$\int_{0}^{\pi/4} \int_{0}^{sin4\theta}\int_0^{3r} r\ dz\ dr\ d\theta$$
please also explain why coz I also do not understand why is the jacobian multiplied at all
Area with Polar Form When you find the area under a curve in rectangular form, you’re measuring the amount of space between the curve and the x-axis. To find the area, you’re essentially dividing the curve into an infinite number of infinitely thin rectangles. By finding the sum of the areas of all the rectangles, you can find the area under the entire curve.
When you find the area under a curve in polar form, you’re finding the area of the space bounded by the curve and two rays that share a common origin. You’re finding the area enclosed by an infinite number of infinitely small sectors.
The formula for the area under a curve in polar form takes this difference into account.
Finding the area under a polar curve can be a bit more complicated than finding the area under a rectangular curve. First, determine what sort of curve the equation represents. Then sketch the curve and figure out what angles of θ form the boundaries of the area you plan to integrate (be very careful with this step!). Finally, set up the integrals and solve.
For volume Same as above procedure
∫π/40∫sin4θ0∫3r0r dz dr dθ