Evaluate the integral by interpreting it in terms of areas. In other words, draw a picture of the region the integral represents, and find the area using high school geometry. ∫90|9x−8|dx=

Question

 Evaluate the integral by interpreting it in terms of areas. In other words, draw a picture of the region the integral represents, and find the area using high school geometry.

∫9 to 0 of |9x−8|dx=

Answer 1

Giulia

If you graph this in Desmos or on a graphing calculator you will see the key points.

Touches the x axis when 9x - 8 = 0, ie x = 8/9

When x = 0, y = 8.

So you can split the integral into two areas. From x = 0 to x = 8/9, will be the area of a right triangle

base 8/9, height 8.

From x = 8/9 to x = 9, will be the area of a right triangle. Base ( 9 - 8/9). When x = 9, y = 73 so height will be 73.

Find the areas of these two right triangles and add together to get your answer.

Mike