Question

I need help evaluating integrals

Answer 1

b) 360

c) -108

d) 72

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

`\int\limits^(30)_(0) {f(x)} \, dx = 360`

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

`\int\limits^(42)_(30) {f(x)} \, dx = -108`

d)
`\int\limits^(54)_(0) {f(x)} \, dx =`
`\int\limits^(30)_(0) {f(x)} \, dx`
`+ \int\limits^(42)_(30) {f(x)} \, dx + \int\limits^(54)_(42) {f(x)} \, dx`

`\int\limits^(54)_(0) {f(x)} \, dx =` 360-108 + (-18 - 12)(12)/2 = 360-108-180 = 72