Questions: ∫ from 4 to 6 h(x) dx = □ ∫ from -5 to 6 h(x) dx = □ ∫ from -2 to 4 h(x) dx = □ ∫ from -5 to 4 h(x) dx = □ ∫ from 4 to 5 h(x) dx = □ ∫ from -2 to 2 h(x) dx = □ ∫ from -4 to 5 h(x) dx = □ ∫ from -4 to 4 h(x) dx = □

Solution Steps

The area under the curve from $x=4$ to $x=6$ is a triangle with base 2 and height 1. The area is $(1/2)(2)(1) = 1$.

The area under the curve from $x=-5$ to $x=-4$ is a triangle with base 1 and height 4. Area = (1/2)(1)(4) = 2. The area under the curve from $x=-4$ to $x=-2$ is a rectangle with base 2 and height 4. Area = (2)(4) = 8. The area under the curve from $x=-2$ to $x=0$ is a triangle with base 2 and height 2. Area = (1/2)(2)(2) = 2. The area under the curve from $x=0$ to $x=1$ is a triangle with base 1 and height 1. Area = (1/2)(1)(1) = 1/2 = 0.5. The area under the curve from $x=1$ to $x=2$ is a triangle with base 1 and height 1. Area = (1/2)(1)(1) = 1/2 = 0.5. The area under the curve from $x=2$ to $x=4$ is a triangle with base 2 and height 5, but is below the x-axis. Area = (1/2)(2)(5) = -5. The area under the curve from $x=4$ to $x=6$ is a triangle with base 2 and height 1. Area = (1/2)(2)(1) = 1.

Total Area = $2+8+2 + 0.5+0.5-5+1 = 8.0$

The area under the curve from $x=-2$ to $x=0$ is a triangle with base 2 and height 2. Area = (1/2)(2)(2) = 2. The area under the curve from $x=0$ to $x=1$ is a triangle with base 1 and height 1. Area = (1/2)(1)(1) = 1/2 = 0.5. The area under the curve from $x=1$ to $x=2$ is a triangle with base 1 and height 1. Area = (1/2)(1)(1) = 1/2 = 0.5. The area under the curve from $x=2$ to $x=4$ is a triangle with base 2 and height 5, but is below the x-axis. Area = (1/2)(2)(5) = -5. Total Area = $2 + 0.5 + 0.5 - 5 = -2$

Final Answer: