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

1 Answer
Nov 15, 2016

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