Question

35,453 Learners found this answer helpful

1. calculate the definite integrals by referring to the following figure.

area ( a = 1.408 )
area ( b = 2.475 )
area ( c = 5.333 )
area ( d = 1.792 )
(1) ( int_{a}^{b} f(x) d x )
(2) ( int_{a}^{0} f(x) d x )
(3) ( int_{c}^{0} f(x) d x )
(4) ( int_{b}^{d} f(x) d x )

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) \(3.541\)