question find the total area between the graph of the function f(x)=1 - |x - 1|, graphed below, and the x…
question find the total area between the graph of the function f(x)=1 - |x - 1|, graphed below, and the x - axis over the interval -3,3.
Answer
Explanation:
Step1: Split the absolute - value function
The function (y = 1-\vert x - 1\vert) can be written as a piece - wise function. When (x-1\geq0) (i.e., (x\geq1)), (y = 1-(x - 1)=2 - x). When (x - 1<0) (i.e., (x<1)), (y = 1-(1 - x)=x).
Step2: Calculate the area using integration
We split the integral over the interval ([-3,3]) into two parts: (\int_{-3}^{1}x dx+\int_{1}^{3}(2 - x)dx). First, calculate (\int_{-3}^{1}x dx). Using the power rule (\int x^n dx=\frac{x^{n + 1}}{n+1}+C(n\neq - 1)), we have (\left[\frac{x^{2}}{2}\right]{-3}^{1}=\frac{1^{2}}{2}-\frac{(-3)^{2}}{2}=\frac{1}{2}-\frac{9}{2}=-4). But we want the area, so we take the absolute - value (\vert - 4\vert = 4). Second, calculate (\int{1}^{3}(2 - x)dx=\left[2x-\frac{x^{2}}{2}\right]_{1}^{3}=(2\times3-\frac{3^{2}}{2})-(2\times1-\frac{1^{2}}{2})=(6-\frac{9}{2})-(2 - \frac{1}{2})=( \frac{12 - 9}{2})-( \frac{4 - 1}{2})=\frac{3}{2}-\frac{3}{2}=0). The total area (A = 4+0 = 4).
Answer:
4