Question

prove area of traingle is 1/2 base × height using integration ​

Answer 1

I am not sure what level are you and what level of answer you want.

as question is not tagged with trignometry or calculus or any such fancy words i will try to keep it simple  

ok i will start with the assumption that area of square with side x is x2 sq units

// if you dont agree with my assumption no need to proceed

using that you can easily prove that area of a rectangle with length l  and  width b is  lb sq units

// how??

just divide the rectangle into small squares and you will reach the result eventually :P

now what can we do with triangle

for easy case :P lets try to find area of a right triangle with base b and height h , and we need to calculate its area.

To find that lets consider a rectangle of lenth h and width b

as we know area of rectangle is bh and i will not prove it but diagonal of rectangle divides it into two equal areas and voilla the half area is nothing but the area of right angled triangle we are interested in .

So i can claim that area of right angled triangle with base b and height h is  

half the area of rectangle and hence 12bh

so lets try to see a scalene triangle with no special property like right angle.

let the base be b and corresponding altitude (height) be h

but wait we can divide it into two right triangles as shown here  

now we can say area of complete triangle is nothing but sum of areas of two right triangles one with base x and one with base y // x+y = b

so total area = 12xh+12yh

                     = 12h(x+y)

                     = 12h(b)

                     = 12hb

which is nothing but 12(base)(height)

// hope it makes sense