question the piecewise function f(x) is graph...
question the piecewise function f(x) is graphed below. use geometric formulas to evaluate the following definite integral. ∫₀¹⁰ f(x)dx submit your answer as an exact value. provide your answer below:
Answer
# Explanation:
## Step1: Divide the region
Divide the area under the curve from \(x = 0\) to \(x=10\) into geometric - shapes. We have a triangle from \(x = 0\) to \(x = 3\), a triangle from \(x = 3\) to \(x = 4\), and a rectangle from \(x = 4\) to \(x = 8\), and a triangle from \(x = 8\) to \(x = 10\).
## Step2: Calculate area of first triangle
The first triangle from \(x = 0\) to \(x = 3\) has base \(b_1=3\) and height \(h_1 = 6\). The area of a triangle is \(A=\frac{1}{2}bh\). So \(A_1=\frac{1}{2}\times3\times6 = 9\). But since it is below the \(x\) - axis, \(A_1=- 9\).
## Step3: Calculate area of second triangle
The second triangle from \(x = 3\) to \(x = 4\) has base \(b_2 = 1\) and height \(h_2=6\). So \(A_2=\frac{1}{2}\times1\times6 = 3\).
## Step4: Calculate area of rectangle
The rectangle from \(x = 4\) to \(x = 8\) has base \(b_3=4\) and height \(h_3 = 1\). The area of a rectangle is \(A = bh\), so \(A_3=4\times1=4\).
## Step5: Calculate area of third triangle
The third triangle from \(x = 8\) to \(x = 10\) has base \(b_4 = 2\) and height \(h_4=1\). So \(A_4=\frac{1}{2}\times2\times1 = 1\).
## Step6: Sum up the areas
\(\int_{0}^{10}f(x)dx=A_1 + A_2+A_3+A_4=-9 + 3+4 + 1=-1\).
# Answer:
\(-1\)