Question

Find the area of the region between the curves y = sin(πx), y = x^2 −x, and x = 2. The region involves all three curves.

Answer 1

`(4)/(\pi)+1`

Explanation:

Find intersection points:

`x^2-x=2\\x^2-x-2=0\\(x-2)(x+1)=0`

x = 2 and x = -1 are intersection points.

`x^2-x=sin(\pi x)`

x = 0 and x = 1 are intersection points.

So,

`\int\limits^1_0 (sin(\pi x)-(x^2-x))dx + \int\limits^2_1 ((x^2-x)-sin(\pi x))dx=\\\\=(-(1)/(\pi)cox(\pi x)-(1)/(3)x^3 +(1)/(2)x^2)|\limits^1_0+((1)/(3)x^3-(1)/(2)x^2)+(1)/(\pi)cox(\pi x)|\limits^2_1=\\\\=(4)/(\pi)+1`