5.1 Areas Between Curves

Areas Between Curves: Integrating with Respect to X

Given two curves f(x) and g(x), the area between the two is calculated by: $A=\int_{a}^{b}\!\left\lbrack f\left(x\right)-g\left(x\right)\right\rbrack\,dx$

• where a and b are two points along the x-axis to which the area is bound between

• remembered as solving for the integral of the top function subtracted by the bottom function

Areas Between Curves: Integrating with Respect to Y

Given two curves f(y) and g(y), the area between the two is calculated by: $A=\int_{c}^{d}\!\left\lbrack f\left(y\right)-g\left(y\right)\right\rbrack\,dy$

• where c and d are two points along the y-axis to which the area is bound between

• remembered as solving for the integral of the right function subtracted by the left function

Reminders for solving:

1. Helps to draw a picture (even if its just a rough sketch)

2. If not given the intersection points of the two lines:

   • solve by setting the two functions equal to each other and solving

3. If explicitly indicates wether to integrate with a specific axis:

   • make sure both functions are in y= (if solving in terms of x) or x= (if solving in terms of y)

   • make sure both functions are in the same terms