Answer:
b) 360
c) -108
d) 72
Step-by-step explanation:
b) from 0 to 30, we have a 2 trapezoids' areas to solve.
trapezoid 1: from 0 to 12
A1 = 144, given by part a
A2 = (6+18)(18)/2 = 216
144 + 216 = 360
[tex]\int\limits^{30}_{0} {f(x)} \, dx = 360[/tex]
c) integral from 30 to 42 is just a triangle whose area will be negative since the y is negative
A = xy/2 = 12(-18)/2 = -108
[tex]\int\limits^{42}_{30} {f(x)} \, dx = -108[/tex]
d) [tex]\int\limits^{54}_{0} {f(x)} \, dx =[/tex][tex]\int\limits^{30}_{0} {f(x)} \, dx[/tex][tex]+ \int\limits^{42}_{30} {f(x)} \, dx + \int\limits^{54}_{42} {f(x)} \, dx[/tex]
[tex]\int\limits^{54}_{0} {f(x)} \, dx =[/tex] 360-108 + (-18 - 12)(12)/2 = 360-108-180 = 72
