To find capacitance between two conductors from first principles, a test charge of Q is assumed to be moved from one body to the other. The energy spent in doing this per unit charge gives V. Q/C gives the capacitance.

For parallel plate capacitance, the electric field between the plates are assumed parallel. Most text books assume infinite plate dimensions for derivation so as to get a perfectly parallel electric field everywhere between plates. In practice, if the distance between the plates are much smaller than the dimensions of the plates, the parallel electric field assumption still hold good.

For your configuration, the electric field is not parallel due to the non parallel arrangement of the plates and difference in plate lengths. The figure drawn shows a very exaggerated non parallelism; I think. However, if the angle \$\theta\$ is small, the plates can be assumed mostly parallel and the electric field lines between the conductors can be assumed parallel without introducing much error in the result. for mostly parallel plates, the method of breaking it up into smaller capacitors is correct method.

Once the parallel plate condition is assumed, you need to integrate only the overlapping area of the plates; in this case the length of the smaller plate.

For large angles \$theta\$, the result obtained this way will have significant error.

To find capacitance between two conductors from first principles, a test charge of Q is assumed to be moved from one body to the other. The energy spent in doing this per unit charge gives V. Q/C gives the capacitance.

For parallel plate capacitance, the electric field between the plates are assumed parallel. Most text books assume infinite plate dimensions for derivation so as to get a perfectly parallel electric field everywhere between plates. In practice, if the distance between the plates are much smaller than the dimensions of the plates, the parallel electric field assumption still hold good.

For your configuration, the electric field is not parallel due to the non parallel arrangement of the plates and difference in plate lengths. The figure drawn shows a very exaggerated non parallelism; I think. However, if the angle \$\theta\$ is small, the plates can be assumed mostly parallel and the electric field lines between the conductors can be assumed parallel without introducing much error in the result. for mostly parallel plates, the method of breaking it up into smaller capacitors is correct method.

Once the parallel plate condition is assumed, you need to integrate only the overlapping area of the plates; in this case the length of the smaller plate.

For large angles \$\theta\$, the result obtained this way will have significant error. For such cases you need to solve for the actual shape of the electric field and integrate appropriately.