the graph of f is shown. evaluate each integral by interpreting it in terms of areas.\n(a)…

Explanation:

Step1: Analyze integral (a)

The region for $\int_{0}^{18}f(x)dx$ is a trapezoid. The formula for the area of a trapezoid is $A=\frac{1}{2}(b_1 + b_2)h$. Here, $b_1=18$, $b_2 = 36$, $h = 18$. So $A=\frac{1}{2}(18 + 36)\times18=486$.

Step2: Analyze integral (b)

The region for $\int_{0}^{45}f(x)dx$ is composed of the trapezoid from $x = 0$ to $x=18$ and a rectangle from $x = 18$ to $x = 36$ and a triangle from $x=36$ to $x = 45$. The area of the trapezoid is $486$ (from step 1). The area of the rectangle with length $18$ and width $36$ is $18\times36 = 648$. The triangle has base $9$ and height $36$, so its area is $\frac{1}{2}\times9\times36=162$. The total area is $486+648 + 162=1296$.

Step3: Analyze integral (c)

The region for $\int_{45}^{63}f(x)dx$ is a triangle with base $18$ and height $- 36$ (negative since it's below the $x$-axis). The area is $\frac{1}{2}\times18\times(-36)= - 324$.

Step4: Analyze integral (d)

The region for $\int_{0}^{81}f(x)dx$ is the sum of the areas from previous parts. $\int_{0}^{45}f(x)dx=1296$ and $\int_{45}^{63}f(x)dx=-324$. For the part from $x = 63$ to $x = 81$, it's a triangle with base $18$ and height $18$, so its area is $\frac{1}{2}\times18\times18 = 162$. The total is $1296-324 + 162=1134$.

Answer:

(a) $486$ (b) $1296$ (c) $-324$ (d) $1134$