3. calculate the definite integrals by referring to the following figure.\narea ( a = 1.408 )\narea ( b =…

Explanation:

Step1: Recall the property of definite - integral and area

If (y = f(x)), then (\int_{a}^{b}f(x)dx) is equal to the net - area (area above the (x) - axis minus area below the (x) - axis) between the curve (y = f(x)), the (x) - axis, and (x=a,x = b).

(1) Calculate (\int_{a}^{b}f(x)dx)

The region from (x = a) to (x = b) is above the (x) - axis. By the property of definite integral and area, (\int_{a}^{b}f(x)dx=\text{Area }A). So, (\int_{a}^{b}f(x)dx = 1.408)

(2) Calculate (\int_{a}^{0}f(x)dx)

The region from (x=a) to (x = 0) consists of two parts: from (x=a) to (x = b) (above the (x) - axis) and from (x = b) to (x = 0) (below the (x) - axis). By the property (\int_{a}^{0}f(x)dx=\int_{a}^{b}f(x)dx+\int_{b}^{0}f(x)dx). Since (\int_{b}^{0}f(x)dx=-\text{Area }B) (because the region is below the (x) - axis) and (\int_{a}^{b}f(x)dx=\text{Area }A). (\int_{a}^{0}f(x)dx=\text{Area }A-\text{Area }B) Substitute (\text{Area }A = 1.408) and (\text{Area }B=2.475) (\int_{a}^{0}f(x)dx=1.408 - 2.475=-1.067)

(3) Calculate (\int_{c}^{0}f(x)dx)

The region from (x = c) to (x = 0) is above the (x) - axis. By the property of definite integral and area, (\int_{c}^{0}f(x)dx=\text{Area }C) So, (\int_{c}^{0}f(x)dx = 5.333)

(4) Calculate (\int_{b}^{d}f(x)dx)

The region from (x = b) to (x = d) consists of two parts: from (x = b) to (x = c) (above the (x) - axis) and from (x = c) to (x = d) (below the (x) - axis). By the property (\int_{b}^{d}f(x)dx=\int_{b}^{c}f(x)dx+\int_{c}^{d}f(x)dx). Since (\int_{b}^{c}f(x)dx=\text{Area }C) and (\int_{c}^{d}f(x)dx=-\text{Area }D) (because the region is below the (x) - axis) (\int_{b}^{d}f(x)dx=\text{Area }C-\text{Area }D) Substitute (\text{Area }C = 5.333) and (\text{Area }D = 1.792) (\int_{b}^{d}f(x)dx=5.333-1.792 = 3.541)

Answer:

(1) (1.408) (2) (-1.067) (3) (5.333) (4) (3.541)