page+376+number+33+(Rachel)

For these problems, you can either count the boxes under the curve or use formulas from geometry. a) ∫ f(x)dx from x=0 to x=2 Split into rectangle and triangle. Area triangle = 1/2(bh). Base = 2, height = 2 => area = 2 Area rectangle = lw. Length = 2, width = 2 => area = 2 Total area: 2+2=4

b) ∫ f(x)dx from x=0 to x=5 Split into 2 rectangles and 2 triangles. Triangle 1 = connect points (3,3), (3,0) and (5,0) => area = .5(bh) => .5(2*3) = 3 Triangle 2 = connect points (0,1), (2,1) and (2,3) => area = .5(bh) => .5(2*2) = 2 Rectangle 1 = connect points (0,0), (3,0), (0,1) and (3,1) => area = lw => 3*1 = 3 Rectangle 2 = connect points (2,3), (3,3), (2,1) and (3,1) => area = lw => 1*2 = 2 Total area = 3+2+3+2 = 10

c) ∫ f(x)dx from x=5 to x=7 Area of triangle = .5(bh) => .5(2*-3) = -3

d) ∫ f(x)dx from x=0 to x=9 x=0 to x=5 => area = 10 x=5 to x=7 => area = -3 x=7 to x=9: Split into triangle and square. Triangle = connect points (7,-3), (7,-2) and (9,-2) Area triangle = .5(bh) = .5(2*1) = -1 Square = connect points (7,0), (9,0), (7,-2) and (9,-2) Area square = -4 Area from x=7 to x=9 = -5

Total area from x=0 to x=9: 10+-3+-5 = 2