Question

How to find the area under the curve y=2x+4 from x=0 to x=2 ?

Answer 1

Either find `int_0^2 (2x+4) dx` or use geometry.

Explanation:

Using the integral, we evaluate:

`int_0^2 (2x+4) dx`

Using the Fundamental theorem of Calculus, we get:

`int_0^2 (2x+4) dx = (x^2+4x)]_0^2`

`= [(2)^2+4(2)] - [(0)^2+4(0)]`

`= 12`

Without the integral, we could note that the region is a rectangle with base 2 and height 4, topped by a triangle with base 2 and height 4, so the total area is:

`(2)(4) + (1/2(2)(4)) = 8 + 4 = 12`

Here is a picture of the region:

graph{(y-2x-4)sqrt(2x-x^2)(y) <= 0 [-5, 5, -1, 10]}