Integral calculation using areas

[mech-eng]

Homework Statement

The problem requires interpreting integrals as areas when calculating. $\int_1^2 (1-2x)dx$

Relevant Equations

Are of a triangle is base X height.

Evaluate the integral using the properties of definite integral and interpreting integrals as areas.

$\int_{-1}^2 (1-2x)dx$

I need to see there are two areas and these are the same but one is under x-axis the other is above x-axis so the value of the integral is zero. To see this is difficult to me.

Source: Calculus A Complete Course by Robert A. Adams

Thanks.

 

[LCKurtz]

Evaluate the integral using the properties of definite integral and interpreting integrals as areas.

$\int_{-1}^2 (1-2x)dx$

I need to see there are two areas and these are the same but one is under x-axis the other is above x-axis so the value of the integral is zero. To see this is difficult to me.

Source: Calculus A Complete Course by Robert A. Adams

Thanks.

In the right hand triangle the height is negative because $y$ is negative. That's why the integral for that part gives a negative area. If you want the geometric area use $y_{\text{upper}}-y_{\text{lower}} = 0 -(1-2x)$ in your integrand for the right hand integral.

 

[fresh_42]

One area is oriented: first in direction +x then in direction -y (green), and the other area is oriented: first in direction +x then in direction +y (red), which results in a different sign, because the orientation has changed.
To calculate the area, the absolute values of both integrals have to be added (split att x=1/2), and to calculate the integral, the areas will cancel out to zero.