Question

How do you evaluate the definite integral `int2xdx` from `[0,1]`?

Answer 1

1 square unit

Explanation:

`int_0^1 (2x)dx`

You can solve this many ways:

1. (easiest)Think of it geometrically, as a line `y=2x`, and find the area under the curve from `0` to `1` using the area of a triangle formula

2. (easy)Use a fundamental theorem of calculus, and say that if `f(x)=2x`, and `F(x)` is the antiderivative of `f(x)`, then `int_0^1 (2x)dx=F(1)-F(0)`

3. (hard)Use Riemann sums to find the area under the curve from `0` to `1` using an infinite number of rectangles

Method #1: graph{2x [-2.784, 2.69, -0.523, 2.215]} Area of triangle#=(1/2)(b)(h)# #=(1/2)(1)(2)# #=1# square unit

Method 2:
`int_0^1 (2x)dx`
Let `f(x)=2x`
`F(x)=int2x`
`=x^2+c`
`F(1)-F(0)`
`=1-0`
`=1`square unit