Can you find the area of the gray region? He asked for help

Find the area of the gray region: a semicircle of radius 8 inside a shape with sides 10 and 14. The trick is one radius drawn straight down. Then two trapezoids, two quarter circles, and the answer 160 minus 32 pi. At the end we show why the trapezoid rule works. CHAPTERS 0:00 The post on Threads 0:27 Drawing the figure 1:21 Why it is a trapezoid 1:50 The key line 2:45 The quarter circle 3:07 The trapezoid rule 3:26 The left trapezoid 4:02 The right trapezoid 4:40 Adding the pieces 5:16 Why the rule works 6:28 Back to the question