Simpson's Rule. It states that, sum of first and last ordinates has to be done. Add twice the sum of remaining odd ordinates and four times the sum of remaining even ordinates. Multiply to this total sum by 1/3rd of the common distance between the ordinates which gives the required area.
For regular shapes:
Rectangles - length time breadth.
Circles - pi*r2
Triangles
Learn the Triangle Inequality Theorem. This theorem simply states that the sum of two sides of a triangle must be greater than the third side. If this is true for all three combinations, then you will have a valid triangle. You'll have to go through these combinations one by one to make sure that the triangle is possible. You can also think of the triangle as having the side lengths a, b, and c and the theorem being an inequality, which states: a+b > c, a+c > b, and b+c > a.
The app will do these checks for you.
Herons formula is used: https://www.mathopenref.com/heronsformula.html
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