Teaching integration to kids

First, your task is impossible to teach rigorously. Since these students probably don't have the algebra skills necessary to set the groundwork to success in calculus. But this is what I would do as my lesson plan.

First define what area even means in terms of area of a rectangle. The area of a rectangle is $length*width$ or $a*b$.

Next show areas are additive and show other areas. So the area of a triangle is $\frac{1}{2}a\cdot b$. But WHY? Well explain to them area is additive and show them this is what the area of a triangle has to be, to be consistent with the area of a rectangle. Then provide as many proofs as you want to show that Area of A + Area of B=Area of A & B.

I would also stick to geometric ways of showing most of my ideas.

Also show how rectangle A, $1\:\cdot \:2$, has the same area as rectange B, $\sqrt{2}\cdot \sqrt{2}$.

And as a geometric way to help them understand so you can cut one rectangle to achieve the other rectangle, reinforcing areas can be subtracted and added.

Anyway, once you have them interested in the idea that area that if two things have the same area you can geometrically chop one thing into pieces and form the other thing; then introduces curves. This idea of area being able to cut pieces and rearrange it to form another thing with the same area is totally destroyed with objects with curved boundaries.

For example what does it mean to have an area of $\pi $.Well then teach them that they can find area in terms of rectangles which they already know. Show them they can approximate by making small enough rectangles. Obviously we don't care to much about computations as much as giving them a general understanding, so I wouldn't go overboard with it. I wouldn't introduce limits directly, but make sure they understand the smaller the rectangles the better approximation they get.

Now at this point I finally start talking about curves and introduce some notation. I start with a simple curve and partition it into rectangles.

I say $dx$ represents the width of the tiny rectangle, $f(x)$ is the height, $\int $ is sum of, and $b$ and $a$ is my interval. I am not getting technical whatsoever to what these actually represent! Not defining limits, not going into summations, anti-derivatives or the fundamental theorem.

Show them something like $\int _0^1x^2dx=\frac{1}{3}$ and ask them what $\int _0^13x^2dx$

Finally teach them about how they can approximate areas of some small curves.

I would use desmos online graphing calculator to show them the how to use rectangular area approximations.

This may not answer your question. But it will probably be a good response. DON'T TEACH CALCULUS to kids! I suggest you teach them something more enriching and realistic like paw88789 said. There isn't a need for these kids to learn calculus at such a young age. Calculus can be very confusing and complex, they should wait till they mature a little bit.

I have been selected by my college to teach integration to kids in the age group of 8-12.

Go to who ever selected you to teach calculus and explain to them how unrealistic his or her goal is to accomplish. Is there a reason why they want you to teach calculus to these kids?

I am an engineering major who has finished Calculus 1 and 2 but I have no idea how to teach integration from scratch to kids that small

You shouldn't have an idea on how to teach it. Some of these kids won't know the multiplication table, how are you going to teach them calculus? The answer is you wont.

and at the same time make it fun for them.

Teach them about something they can understand. And that applies to them in nature. Something they can talk about with their friends. Or just be an easy going teacher. Maybe make some math jokes. But stay away from calculus. Not that calculus can't be fun, but they can't understand it or apply it.

I am asked to create a lesson plan, worksheets, manipulatives etc. Any help is appreciated. Parents/Teachers, your ideas will really come handy.

If you really can't avoid teaching them something more useful or passing on the offer. Then teach them about slopes instead. Then show them the bare bones concept of derivatives. But stay away from integration and anti-differentiation. Anti-differentiation is like anti-multiplying or factoring, it requires a lot more skill. Derivatives itself are a huge stretch, but is a whole lot more reasonable of a task than integration. Ideally stay away from calculus altogether.

Most of these kids are not going to be math majors and have no purpose for learning calculus especially at that young of an age.

Define integration as area under a curve.

Use examples to approximate well-known areas (start with line, e.g. area of trapezoid, and make it more complex like semi-circle or half-ellipse or parabola) with rectangles and see the numbers getting closer to what the area is.

You cannot teach analytic integration because they won't be able to find the anti-derivative, so stay in geometric approximations. Not sure you can do much else at that age.

I don't have any specific advice, but I remember hearing about a book that was really good; Calculus for Young People. Maybe you could get in contact with the author as well?